P H O N O N - P H O N O N I N T E R A C T I O N S I N T H E T H E O R Y OF F L U I D S by JOHN ERWIN LOKKEN A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in PHYSICS We accept this thesis as conforming to the standard required from Candidatea for the degree of DOCTOR OF PHILOSOPHY . Members of the Department of Physics T H E U N I V E R S I T Y OF B R I T I S H C O L U M B I A OCTOBER, 1955 ABSTRACT This thesis i s devoted to the effect of phonon-phonon inter-actions on the energy of a non-viscous f l u i d , and hence to i t s specific heat. It extends the work of previous authors by taking into account the terras i n the Hamiltonian of higher than the second order (lowest order) i n the f i e l d variables. It i s shown that the term' of third order in the f i e l d variables, contributing i n second order, and the term of fourth order, contributing i n f i r s t order, may give significant con-tributions i f the theory i s applied to liquid helium II. The phonon energy in this approximation i s linear i n the momentum, = 0 +«t-i)c0^»K- Here, dC0HK i s the contribution of the fourth order term and -%cjf\\k the contribution of the third order term. C'hK i s the value obtained by previous authors by considering the lowest order term only. It i s shown that for liquid helium II the contribution to the energy of the non-linear terms i s smaller than that of the linear terms. In this expression for the energy (l + U i s interpreted as the measured velocity of f i r s t 3ound. Thus the cubic term i n the specific heat i s unchanged i n this approach, the effect of the higher order terms i n the Hamiltonian being to change the velocity of sound. Because the non-linear terms have been found to be small the conclusion has been reached that the terms of higher order i n the tem-perature than the third cannot be attributed to phonons, and that there-., fore this theory i s only valid for liquid helium II below 0.6°K. ACKNOWLEDGEMENTS The research described i n the thesis was carried out with the aid of National Research Council Studentships (1952-53, 1953-54) and Summer Scholarships (1955* 195*0 t o the author. I wish to express my gratitude to Professor F. A. Kaempffer, who supervised this research, for suggesting the problem, for" his continued interest in i t and for the many helpful discussions we have had. Finally, I wish to.express my appreciation to my wife for her constant encouragement, and the shouldering of many of the parental responsibilities that should have been mine. Faculty of Graduate Studies P R O G R A M M E O F T H E Jffittd ©nrl ^Examination for t\}t ]&t$ztz B.Sc. (University of Western Ontario) 1951 M.Sc. (University of Western Ontario) 1952 M O N D A Y , O C T O B E R 17th, 1955 at 3:00 p.m. I N R O O M 303, PHYSICS B U I L D I N G C O M M I T T E E I N C H A R G E DEAN H . F . ANGUS, Chairman F . A . KAEMPFFER C. REID L . E . H . TRAINOR T . E . HULL G . M . VOLKOFF A . D . MOORE G. M . SHRUM G . O. B. DAVIES of of JOHN ERWIN LOKKEN External Examiner—G. K . HORTON University of Alberta THESIS Phonon-Phonon Interactions in the Theory of Fluids. This thesis is devoted to the effect of phonon-phonon interactions on the energy of a non-viscous fluid, and hence to its specific heat. It extends the work of previous authors by taking into account the terms i n the Hamiltonian of higher than the second order (lowest order) i n the field variables. It is shown that the term of third order i n the field variables, contributing in second order, and the term of fourth order, contributing in first order, may give significant contribu-tions if the theory is applied to l iquid helium II. The phonon energy in this approximation is linear in the momen-tum, E = (1 + C L - S) Q / f i K . Here ,#C n ' #K is the contribution of the ft fourth order term and -SC0"#K the contribution of the third order term. CQ'KK is the value obtained by previous authors by considering the lowest order term only. It is shown that for l iquid helium II the contribution to the energy of the non-linear terms is smaller than that of the linear terms. This expression for the energy is used to calculate the specific heat of l iquid helium II below 0.6°K. It is shown that the measured specific heat can be accounted for completely using reasonable values of oC and S • Because the non-linear terms have been found to be small the con-clusion has been reached that while the discrepancy in the cubic term of the specific heat can be accounted for, terms of higher order in the temperature can not be grasped by this approach, and that therefore this theory is only valid for l iquid helium II below O.6 0 K. PUBLICATION Contribution from Phonon-Phonon Interactions to the T 3 Term of the Specific Heat of Hel ium II, Canadian Journal of Physics, 32, 359, 1954. GRADUATE STUDIES Field of Study: Physics Quantum Mechanics J . H . Blackwell Tensor Calculus and Relativity Theory E . Brannen Theory of Noise E . H . T u l l Statistical Mechanics E . Brannen General Theory of Relativity F. A . Kaempffer Quantum Theory of Radiation H . Koppe Physics of the Solid State H . Koppe Theoretical Physics Seminar W . Opechowski Theoretical Nuclear Physics W . Opechowski Electromagnetic Theory J . M . Brown Other Studies: Network Theory A . D . Moore Integral Equations T . E . H u l l TABLE OF CONTENTS CHAPTERS ' PAGE I Introduction and Summary 1 1.1 Historical Review 1 1.2 Program of This Work 5 II Development of the Hamiltonian in Operator Form . . . . 9 2.1 The Classical Hamiltonian 9 2.2 Quantization 10 2.3 Division into Unperturbed and Perturbation Hamiltonians 19 III The Energy Spectrum of Phonons and Rotons 24 3.1 The State Function • 24 3.2 Contribution of the Unperturbed Hamiltonian . . . . 2.7 3.3 Contribution of the Perturbation Hamiltonian. . . . 30 IV Possible Application to Liquid Helium II 49 APPENDIX A Proof of Eq. (11.33) 53 TABLES IV. 1 p , V and i?/p at Various Pressures and Temperatures . . 51 A.l Tabulation of f„ _ .56 TABLE OF CONTENTS (continued) PAGE FIGURE III. 1 Region of Integration of Eq. ( 1 1 1 . 2 6 ) . . 5 3 GRAPH FACING PAGE IV. 1 d/p as a Function of j> at Various Temperatures . . . . 5 0 PAGE REFERENCES 5 7 1 CHAPTER I INTRODUCTION AND SUMMARY Phonon-phonon interactions have been investigated in this thesis c as a prototype of a non-linear f i e l d theory, and for their possible applic-ation to the theory of liquid helium II. The work forme an extension of the quantum hydrodynamics of Kronig and Thellung (9) and Ziman (20) by ' taking into account higher order terms in the Hamiltonian. 1.1 Historical Review Quantum hydrodynamics is really a f i e l d theoretical approach to the many particle behaviour of a f l u i d , and as such is applicable only when the excitations of the f l u i d - or excitons as H.A. Kramers has called them ( 7 ) - have a wave length which is long compared to the interatomic spacing. Although a rigorous formulation was not found unt i l 1952, the idea of quantum hydrodynamics was introduced in a now famous paper by Landau in 19^ 1 (11). It was in connection, with liquid helium II that he f i r s t attempted to construct a quantum theory of liquids by direct quant-ization of hydrodynamical variables such as the density, the current and the velocity, without explicit reference to interatomic forces. Thus he sought to express the dynamical behaviour of the liquid at temperatures between zero and a few degrees Kelvin in terms of quantized excitations of the motion, and to this end postulated certain commutation relations between the density, the current and the velocity of the microscopic liquid*; these relations reduce in the macroscopic limit to the ordinary * London (15) has questioned the representation of- the velocity by a linear operator; however, Thellung (18) has shown that except for a minus eign a l l of Landau's commutation rules follow from the more rigor-ous treatment developed Independently by Ito (4), Thellung (18) and Ziman (20). 2 hydrodynamical equations in operator form. He assumed that the Hamilton-ian density was given by the sum of the kinetic energy and the internal energy per unit volume. The longitudinal excitations, due to potential motion of the liquid, called "phoneme", were found to have an energy proportional to the momentum; that i s , E R = C«-ftK c,p , c, being the velocity of f i r s t sound, K the propogation constant and p=-fcK the momentum of the phonon. This leads to a specific heat proportional to T 3. While this i s of the correct power in T at low enough temperatures, i t has been shown by H.C. Kramers, Wasscher and Gorter (8) to be too small in the case of liquid helium II. Except for the special case of irrotational motion, the compon-ents of the velocity do not commute, the commutator of the components of the velocity containing a component of the vorti c i t y . Landau then assumed from the analogy with angular momentum** that the vortex motion of the f l u i d must be quantized. Furthermore, for the existence of super-f l u i d i t y the ground state of the vortex motion must l i e above that of the potential motion; hence he conjectured that the energy of an excitation of the vortex spectrum, which he called a "roton", measured on the energy scale on which potential motion is reckoned as zero, may be given by an expression of the form. E r = * + £ » where the effective mass of the roton JJ and the rest energy A are constants **• The analogy is not complete, however, since for a complete analogy the commutator would equal a component of the velocity rather than the vorticity. adjusted to f i t the experimental specific heat curve: This' completed a phenomenological description of liquid helium II ex-plaining the experimental results to that time. Later, however, Peshkov's (16) measurements of second sound velocities indicated a marked increase in the value of// as the. temper-ature decreases, whereupon Landau modified the roton spectrum so that the rotons were clustered about one. value of momentum, p„ ; i.e. (Ml) where now In the same paper he abandonned his previous concept of rotons and instead considered them as being longitudinal excitations of short wave length. Peshkov's measurements were then used to plot an energy spectrum for longitudinal excitations that was nearly linear i n the mom-entum for small momenta, corresponding to phonons, but quadratic near P« corresponding to rotons. In this way agreement with experiment was ob-tained both for specific heat and for second sound velocities. On the assumption that phonons and rotons are the same type of excitation but of different wave length,Landau and Khalatnikov (13) showed that the energy of a phonon* of small momentum is of the form E K = c c ( p - r P J ) , where Y = 2.8 x 10 S 7 gr* cm^iec* . Temperly (17, §9 page 505)» however, ha8 pointed out that this value of ¥ is somewhat less than one would 4 expect from other considerations. Landau and Khalatnikov i n the same paper.calculated the scatter-ing cross sections for interactions between excltona in connection with the evaluation of the coefficient of viscosity of liquid helium II. The Hamiltonian proposed by Landau (11) oontains terms from which the cross section for' the scattering of phonons by phonons was computed.) i n order to keep the cross sections f i n i t e they used the non-linear energy spectrum mentioned in the previous paragraph. Since at that time there was no quantum mechanical Hamiltonian known that included vortex motion, the rotons were considered as small spheres for the purpose of evaluating the cross sections for the scattering of phonons by rotons and rotons by rotons. The coefficient of viscosity calculated using these cross sections, and thus attributable to the excitons, was in good agreement with experi-ment. A great advance in quantum hydrodynamics was made in 1952 by Kronig and Thellung (9) when they considered the irrotational motion of a non-viscous f l u i d from a f i e l d theoretical point of view. They obtained the classical Hamiltonian, which was of the form assumed by Landau in 19^1, starting from Bernoulli's equation and the equation of continuity. Follow-ing Landau they expanded.the Hamiltonian in a power series i n the canon-i c a l f i e l d variables p (velocity potential) and p (density of the liquid) and then quantised using the usual f i e l d theoretical methods. The term of-lowest order i n the variables (second order) leads to the T term in the specific heat previously obtained by Landau. No attempt was made to evaluate the contribution from higher order terms. Sometime later Thellung (18), Ito (4) and Ziman (20) independ-ently derived a classical Hamiltonian describing both irrotational and rotational motion of an ideal non-viscous f l u i d . On expanding the Hamiltonian in terms of increasing order in the f i e l d variables the phonon terms previously obtained by Kronlg and TheHung result, as well as terms containing only the roton variables and terms containing both phonon and roton variables. Only Ziman was able to obtain any eigen values; and at that, he found them rigorously only for states with no rotons and states with one roton, neglecting the contribution from phonon-roton interactions. In order to keep the energy f i n i t e he found i t necessary to take into account the "graininess" of the underlying f l u i d by the introduction of an upper limit to the propogation vectors. The energy spectrum obtained then was of the form postulated by Landau in 19^1, with For purposes of visualization of the roton states in terms of observable movements of the f l u i d , Ziman used cylindrical coordinates and obtained axially symetric solutions to the wave equation. He found, as expected, that the vortex motion i s quantized and that a roton corres-ponds to what one would c a l l , classically, a steady rotational motion of the f l u i d , capable of moving as a "vortex" through the volume. This motion, however, since i t is quantized can not persist as a steady state with the vorti c i t y a r b i t r a r i l y Bmall but non-zero everywhere. 1.2 Program of This Work It was the purpose of the work reported in this thesis to extend the theory of Kronig and Thellung (9) by taking into account- the inter-action terms of the general Hamiltonian as set forth by Ziman (20). For this investigation the Hamiltonian was expressed i n terms of annihilation and creation operators, and arranged into terms of ascending order in the 6 operators, with the annihilation operators operating before the creation operators. In order to get a system with 3N degrees of freedom, i t was necessary to introduce the "graininess" of the underlying fluid. The ad-vantage of this procedure is that i t introduces an upper limit to the spec-trum, without which the contribution to the energy of terms of order three and higher in the field variables would be infinite. The Hamiltonian in operator form is divided into two parts: the terms that contribute to the energy in first order only, which form the un-perturbed Hamiltonian, )V0 , and the terms that contribute in a l l other orders, which form the perturbation Hamiltonian, . It is shown in Chap-ter III that the unperturbed Hamiltonian gives a contribution to the energy of the phonons which is predominantly linear in the momentum. The major linear contribution comes from the terms of order two in the field variables but there is also a first order contribution from the terms of order four, six, etc. The phonon-phonon interactions described by the perturbation Hamiltonian give a second order contribution linear in the momentum and negative. The main contribution to the energy from phonon-phonon. inter-actions arises from the terms of order three and four in the field vari-ables, under the assumption that the series for the internal energy per unit volume converges sufficiently fast that only the first three terms need be considered. It was previously pointed out by the author (14) that the term of order four in the field variables gives a linear contribution, to the phonon energy and could explain the discrepancy between the experi-mental (8) and theoretical specific heat of liquid helium II below 0.6°K. It has since been pointed out by Horton (23), however, that in view of Prohlich's theory of superconductivity (21,22) the perturbations do not : affect the observed specific heat, but rather the quantity to be interpreted 7 as the velocity of sound. Phonon-roton interactions make a negligible contribution to the phonon energy. They give rise to a cubic term in the phonon energy, but i t is of the wrong sign to agree with Landau; i.a., i t is positive whereas Landau and Khalatnikov obtained a negative cubic term using Landau's energy spectrum for excitons (12). Moreover, the magnitude of y is less, being of the order 103* g."1 cm.-2- sec.Z for helium II at 1°K compared to 2.8 x lo'7g.~E cnu2 sec.2, from Landau's theory. They also contribute a linear term to the roton energy of the wrong sign to agree with experi- ' ment. The theory has been applied in Chapter IV to the case of!liquid helium II. The linear terms in the phonon momentum give rise to e T^ term in the specific heat. It has b9en assumed that the only significant con-tribution to the linear term in the energy - and thus to the T3 term in the specific heat - is from terms of fourth order and less in the field variables. Unfortunately i t is not yet possible to decide experimentally whether or not this assumption is justified, for the experimental results from which the coefficients of the third and fourth order phonon terms, can be calculated do not extend below 1°K. However, higher order terms may be taken into account quite easily i f i t should be necessary since they are included in the theoretical expression. Extension of the experi-mental values below 1°K would be very desirable. The non-linear terms in the energy contribute terms of higher power than the third in the temperature to the specific heat; however, these may be neglected in comparison with the third order term. 8 The conclusion has been reached that the perturbation terms affect the observed velocity of sound and that therefore they do not account for any discrepancy in the measured T^ term in the specific heat. The terms of higher order in the temperature are too small to be of im-portance. It would appear that while Ziman's theory gives a first approx-imation to the roton energy spectrum, the higher order corrections have the wrong sign*. That the proper corrections are not obtained, however, is perhaps to be expected since the momentum p 0 about which the rotons cluster in Landau's 1947 theory corresponds to a phonon wave length less than the average interatomic spacing. One therefore would, hardly expect a theory based on a continuous fluid, and which takes the "graininess" of th9 fluid into account only through a cut off in the spectrum, to give such an energy spectrum except through a fortuitous accident. It there-, fore seems unlikely that a theory of this type will be adequate at temper-atures above about 0.6°K, where the specific heat departs from a T^ form. Therefore the emphasis in this work has been on the phonon spectrum, which should give reliable results for temperatures below 0.6°K in the case of liquid helium II. •Recent work by Horie and Osaka (3) shows that phonon-roton inter-actions contribute essentially to the roton spectrum, and that Ziman's perturbation approach is incorrect. The phonon spectrum Is not affected by this. 9 CHAPTER II DEVELOPMENT OF THE HAMILTONIAN IN OPERATOR FORM In this chapter we discuss the Hamiltonian that w i l l be used in the following chapter to evaluate the energy. The f i r s t section and most of the second i s a review of work carried out principally by Ziman (20), and is included for completeness. The division of the Hamiltonian into unperturbed and perturbation parts in the manner described is due to the author, as is the operator expression obtained for the general term of order s in the f i e l d variables. The writer has shown that the roton Hamiltonian i s infinite i f a l l the terms are taken into account, and therefore only the one present in an incompressible f l u i d , and classically the most important, i s retained. 2.1 The Classical Hamiltonian .Ziman (20), Thellung (18) and Ito (4) have shown that the classical motion of a f l u i d , including both rotational and irrotational it motion, can be described by the Hamiltonian'' X * J/jfH txAy4z, (n.l) where the integral' extends over the volume V of the f l u i d , and the Hamiltonian density H is given by where % The notation of Ziman w i l l be used throughout this thesis except where otherwise noted. 10 is the compressionel energy per unit mass of the fluid, and is the velocity. <p and p , the velocity potential and density of the fluid, are canonical variables describing the irrotational (potential) motion of the fluidj the scalar fields y and <r are canonical variables describing the rotational (vortex) motion of the fluid. Equation (II.A) is related to the usual expression for the velocity by the Clebsch transformation (20, 10 page 248). 2.2 Quantization As usual the transition to quantum mechanics is made by re-quiring that the operators corresponding to the field variables f> and ys and the corresponding conjugate momenta p and «- obey the com-mutation relations with a l l other commutators being zero. Ziman showed that the Hamiltonian could be simplified by the transformation to the complex field operator defined by ¥ satisfies the commutation relations The Hamiltonian density obtained thus is, in Hermitian form, 111.7) 11 y3 W(p) can be expanded as 9) classical where c 2^ ^ is the square of the Avelocity of f i r s t sound; for the operator j>~l a similar series i s obtained i 0 - i - i n + ^ - ' - . - L - l z g +lez(£ .' & M ) P ' A <*» A A This expansion is reasonable because classically at.temperatures near absolute zero, and averaging over.volumes large compared to the inter-atomic spacing one has always \/> -/>ol « f» ; i t is shown in §2.3, however, that application of the series (II.10) appears to lead to a divergency in the quantum mechanical treatment used. Arranging the Hamiltonian density in terms of increasing order in the f i e l d variables f , p , f and If one obtains H» • f where (20) Jo ' The terms of higher order than the fourth are due to the expansion of 12 jiVJ(p) and p"x in power series in p -p0. The terms of the Hamiltonian corresponding to (11.11) are ob-tained by integrating over the period volume V= L « . The integration is facilitated by .resolving the variables into Fourier components in the period volume; that is, new operators are defined by the relations which obey the commutation rules cn.ia) (n.i3) with a l l other terms commuting. The sums are to be taken over a l l values —A of the propogation vector K such that -* Z Tf "* -» where '-O'-IJ L 0 i ^ , L 013 are the vector dimensions of the period volume, and the ni. are positive or negative integers, but not zero (20). Substituting (11.12) into (11.11) and integrating we obtain the total Hamiltonian H = with (LT.tfa) H'» - V « 11 [ -1 Tf;S; Fit R ^ F K , • f , ( f *) R i j R- ft.- -13 <4 . fco^A - » 7 b -br b*> - b A b%bt+Lf b% b- b* ) R _ > . . . . + - i i i i i j -A i, -h -J 3 i , -e, -h J?j -V K, K L. J J v »ap )^ K, Ki K3 * K3 +• • fcj K, Kt K4' (JI.I5cJ) We now introduce phonon annihilation and creation operators and a_*by the transformation ( 3 a n d * obey the commutation rules with a l l other terms commuting. As Kronig and TheHung have shown this transformation diagonalizes , and as will be shown later, certain parts of the even order terms of (11.15)• On substituting (II.16) into (11.15) «« obtain "3 •*)> Cn.l8a) X (<J - < ^ + 1| H O f j t ? i V V , + . x ( a * -d^Hat -at)...(^ - f l t ) + */*,*»«•...* <<C ~aJ X 4- f l.t -)•••(** 5**> (nied) where h, and R£ are unit vectors in the directions K, and , and ° - and Z are defined by It i s convenient for what follows to expand (11.18) and use the commutation rules (IX.12) and (11.17) to interchange the order of the operators so that the annihilation operators operate before the creation operators*. One then obtains well ordered expressions for Jr , rt and i r° K, K2 15 * i *5 & The general case of )/^ is discussed later. The quantities f , jj and A In (11.20) are defined by The summations here would be infinite for an ideal fluid; however, in any real fluid excitations with wave lengths of the order of the average ( V \^* -jjl , will not be propogated without some attenu-ation. In fact, i t is unlikely that any can be propogated with wave lengths much shorter than . Thus the summations must have an upper limit not much greater than that corresponding to a wavelength . ("jv)^ • Whether or not there should be a weighting factor, and i f so what its form should be, is not known. Ziman has made the suggestion, similar to that used in the Debye theory of specific heats for solids, that since the J N coordinates of the system of N particles depend only oh p , p must have com-ponents. With this assumption the upper limit is obtained by equating 1 6 the number points i n wave-number space inside a sphere of radius m^ax to JN,thus The corresponding minimum wave-length is ft (1.24) (E.25) so i t is of the order of magnitude one would expect. (11.24) gives s I.S7 x\0 Cm"1 for helium, where m is the mass of the particles constituting the f l u i d . (11.26) has been used throughout this work. On substituting (11.26) into (II.21), (11.22) and (11.23) we obtain = i. id = 9.28 x I O " ' 6 e fgs for helium, 4 3 for helium, We go on now to the general term^'" of order in the f i e l d variables. Let us define n by (1.27) (H.2 7a) (D.28) (£29) (n.30) 17 While (Redoes not occur in one term of the summations in V6**, a l l terms obtained when multiplying i t out are contained among the terms of For small values of p we have ftl* a* °\ «* ** -*aiaiaiai+K «i *» - 4"« v«<<<+ • l e w - ' - * • 'Lwv'i +>-From (11.31) i t would appear that ft could be written in the form A rigorous proof of this without the restriction that the have the same subscript has been relegated to Appendix A. After the calculations carried out in the Appendix one obtains the more general result 2 * and 18 with fP^ being given by the partial difference equation subject to the boundary conditions 0,0 ~ 1 i fp,^ ~ (IT.36) Limited attempts have been made to find a solution of (11.35) in closed form but without success. Since the only important use of is for small values of p with q = 0, 1, a table built up using the di f f e r -ence equation i s adequate and therefore serious attempts at obtaining a solution were not f e l t to be justifiable. From Table A.l page J 6for small values of p and q, i t is evident that for p up to five at least. But since (11.37) is consistent with (11.35) and (II.J6) i t i s li k e l y valid for a l l p. On substituting (11.53) and (11.3*0 into (II.18d) we obtain for the term of order s in the Hamiltonian t?)0 -H / | .3 -5". . - tep- | ) 19 /? ^ ( - ^ y y - ^ — 5 2 2 2 ^ . . . | ^ y [CiW,) • fi, */ 4) £ b : b.. + (ft- /,)i r * L + + ) fl W h^A?* \ ^ i r 5 • • • S K ^ , V 2 f - ' iy-" ZV * 1 W^T! (f&/Mr (T^FT r). 11 • • • ? s - « , ft,**, t . . * 2.J Dlvlelon Into Unperturbed and Perturbation Hamlltonlans Let us look in more detail at V F T | , (II.20c). The state function 2 0 discussed in ^ J . l (Chapter III) diagonalizes the zero point energy term (the term independent of the flj* and a£ ) ; and as well, the second order term i n the tfj*^involving one annihilation and one creation operator. The remaining terms of Becond order in the 5 are not diagonalized by (II . 3 8 ) . hut there are terms of second order in the bp !J that are diago-nalized. In addition we note that there are terms of fourth order in the a^'s and in the l**>s that are diagonal, viz., 0.j> a^fl-.fl- , ty^f-tyf The remaining terms do not contribute in f i r s t order with the state function (11.28). The terms contributing to the energy in f i r s t order do not con-tribute in any other order. 1 Because there are terms which contribute to the energy in f i r s t order exclusively with the remainder not contributing in f i r s t order, the total Hamiltonian has been divided into an unperturbed Hamiltonian, and a perturbation Hamiltonian,}^ . ty0 may contain terms of every even order in the annihilation and creation operators, and hence from every even ordered term in the f i e l d variables. The perturbation Hamiltonian, % l , may contain terms of every order in the operators except zero order. For convenience i n writing, *H0 and V( are divided into groups of terms involving phonon operators only, terms involving roton operators only and mixed,or interaction terms.involving both phonon and roton oper-ators. These are labelled with the subscripts p, r and pr respectively, thus, A superscript i s used to indicate the power in the f i e l d variables to which the term belongs; for instance, specifies the terms of the un-perturbed Hamiltonian arising from the fourth order term in the f i e l d ( a .39) 21 variables, * a«« involving the phonon operators only. The part of the Hamiltonian (11.20) involving roton operators only is given by (1.40) Setting p - 2 = p and changing suffixes V R becomes By using (11.21), (11.22), (II.2J) and (11.57) (11.41) can be written In the form This expression, however, is infinite, for the ratio of the (p +• l)th to the pth term is I'llfHY. M'" ( 2 p + ( ) r , (XLA3) which for y>0 and large enough p is greater than unity. ^ is certainly 22 greater than zero for there is propogation of sound in a l l real fluids. The i n f i n i t y is due to the divergence of the expression for f~' in the quantum mechanical treatment used. Each term i n the summation is of the form of a phonon zero point energy, term (independent of the phonon distribution). It does not seem likely to the writer that the summations over K interact i n such a way that the expansion of />"' converges, and therefore i t appears that i t s expansion i s not valid i n the. treatment used. As a result of the divergence of ^ "'we have resorted in this thesis to the use of the f i r s t term in the expansion, , classically the most important. The terms of the Hamiltonian are, taking into account only the f i r s t term, ~ , in the expansion of P"' , + '-> (n.4+) a 0 ) 2? = W ( 1 1 1 1 t z u *''-JW<-t,% ^ . (™> where, i 24 CHAPTER III THE ENERGY SPECTRUM OF PHONONS AND ROTONS In this chapter the energy of an ideal, non-viscous fl u i d i s found in terms of phonon and roton gases. It is primarily concerned with the contribution of and ^ / ^ t o the phonon energy. To do this i t is necessary to give some consideration to phonon-roton interactions, as well as to phonon-phonon interactions. The contribution of Jl enters through the contribution of through \H(;f» and j>vm ' !Ktfr gives a "self-energy" like contribution to the roton energy. This contribution changes the value given by the unperturbed roton Hamiltonian i n the opposite way to that required to give agreement with experiment, thus i t does not appear to be valid.* It i s further pointed out that one would not expect this type of theory to give Landau's re-vised roton energy. 3it,y also contributes a cubic term to the energy of each phonon, but i t is much smaller than the linear term from ^Cof . cty^ b contributes a negative linear term to the phonon energy. This, together with the contribution from (contained inOfcf ), changes the magnitude of the phonon energy. It i s shown that for helium II the non-linear terms are not important. J . l The State Function The lowest order term in the Hamiltonian contains phonon var-iables only and is of order two in the f i e l d variables and of order zero and two in the creation and annihilation operators. Taken alone i t gives rise to a specific heat of the form which for the case of * See also Horie and Osaka (5). 2 5 liquid helium II is nearly the experimental value below about 0.6°K -the coefficient being somewhat larger experimentally. Thus in this region perturbations must be small, or at least i f there are several per-turbation terms the sum of them must be small. Higher order terms in the Hamiltonian arise from the expansion of the equation of state in a power series in p-p„ and. from the expansion of f~' in a similar power series, series which classically are certainly rapidly convergent at low enough temperatures. One would expect that quantum mechanically the series would also converge rapidly, especially at very low temperatures; however, as pointed out in §2«3» this is hot the case for pml , which diverges. But its divergence is of the same form as the zero point energy divergence (only in this case i t multiplies the roton Hamiltonian, thereby making the roton energy infinite) and therefore does not necessarily imply that the expansion of the equation of state is invalid quantum mechanically. Furthermore, there are coefficients of the form ( ^ pl. in the equation of state terms that may» easily con-tribute to the convergence of the series; in contrast, the expansion of p'1 does not have factors of the form 7J7 to aid its convergence. The validity of the expansion of the.equation of state must be assumed i f perturbation theory is to be applicable to the present calculation. Because of the assumption that the perturbation is small, the use of a state function which diagonalizes and the roton part of .V**1 of second order in bj* and bj» has been considered adequate for evalu-ating the addition to the energy due to phonon-phonon interactions. 26 Such a state function can be 'described in the following wayt Let the state function of Alf excitonB with propogation vector k be written as vf« cpNg . If the excitons are non-interacting the usual relations hold for the annihilation and creation operators (1°), v i z . K « (ECU) When the excitons do not a l l have the same propogation vector and do not interact the state function may be written as the product (m.2) .For two types of excitons that do not interact among themselves nor with each other the state function may be written as the product of the state functions for each type of exciton In- our case we shall l e t Njfbe the number of phonons with mo-mentum ^ and M{* be the number of rotons with momentum h i . We shall use yVi^ f or the state functions of phonons and 'Ypij* for the state functions of rotons. T* given by (III.3) is the state function of the unperturbed Hamiltonian developed in Chapter II. For ease in writing (III.3) has been written in Dirac notation for much of what follows; that i s . i t has been written in the form where the product of the . f^-* * fnf l s *° be understood unless otherwise stated or made evident in the text. 27 J.2 Contribution of the Unperturbed Hamiltonian We shall designate the eigen value of a term in the Hamiltonian by the subscripts and superscripts carried by that term. For the unper-turbed phonon energy we have-..,., - £o)P = < r i ^ i r ) . ( in.5) Explicitly, Eo,P - c?fiV[l + £j£ifi>^(^T)] + ( l t * ) * c e I K J v V + 5- * y. * The f i r s t term is a constant independent of the number of phonons present and Is thus a type of zero point energy, which would be i n f i n i t e i f there were no upper limit to the phonon spectrum. Henceforth i t w i l l be l e f t out of the equations since i t has a fixed value. The remaining terms are functions of the phonon distribution. We assume in analogy with black body radiation that the phonons obey Bose-Einstein s t a t i s t i c s . This choice is further substantiated by the fact that in solids, for which the Debye theory of specific heats is valid, the phonons obey the same sta t i s t i c s . Hence in order to evaluate Eo,f> we replace the summation over k by an integration and the number of phonons by a Bose-Einstein distribution function, thus where E"|? is the energy of each phonon with momentum between *fi K and •fi (K + d . Before carrying out the integration, however, It is necessary to consider the remainder of the Hamiltonian, for i t has terms linear i n Nft which contribute to the energy B R entering into the distribution function. 28 But even though i t is necessary to consider the remainder of the Hamiltonian before carrying out the integration'exactly, i t is con-venient at this point to show that for most purposes E 0 ( P nay be approx-imated by the term linear in K . As we shall see later this will simplify several calculations. We justify this approximation in part by showing that with o<«i, a necessary condition for X perturbation theory to be valid, i t is consistent by. virtue of the non-linear terms then being negligible for helium II. With the assumption that the equation of state converges rapidly enough that we need to consider only the first term in 4 and in ^ , we obtain for helium II In order to make an order of magnitude calculation one may approximate by £-» = C8f)K in the distribution function, and obtain K K r i * V <1'K (nr.9) Since by assumption e(«lthe first term in (III.10), and hance the second in (III.6), is the only one we need to consider in this application. However, in the denominator of second order perturbation calculations the linear terms may cancel; the non-linear terms will then be important. In addition to F^ pwe require the unperturbed roton energy be-fore considering the perturbation Hamiltonian. . It is given after a 29 simple calculation by C-<*l * J r > ' * (EL/k) since M — = M.v- in an isotropic medium. The fi r s t term of (III.11a) with Mj£ = la the result obtained by Ziman for single roton states. (The assumption that Mj^= does not hold for this case.) Since the complete roton Hamiltonian diverges, (III.11a) is probably at best a crude approximation to the roton energy. Nevertheless, let us assume that the energy of each roton is in reality where J\ and//' are not very different from ^ and// in (III.6a). This assumption is in agreement with Landau's earliest prediction (10) as well as with (III.11a), and is for purposes of this thesis a sufficiently accurate approximation since its only use will be in making order of magnitude calculations. Because of the large minimum energy, the rotons will obey Boltzman statistics, thus (III.11a) becomes i x 50 The integrals in (III.1$) are straightforward, giving J7, J'l -_ 3 V E T - ~ __LfL*L (HI. ft) Remembering that f^7~ l7fe| and JJ - ^ *>H , (Ill.l4)may be written as approximately for helium II below 1°K. Clearly, the f i r s t two terms dominate. Since these arise from the f i r s t term of (II.11a), f,r can be taken as £ftr = i(a*\y}^x for liquid heliu,n u> except in cases where this may cencels for instance, when energy differ-ences are taken (III.16) may cancel with a similar expression. 3.3 Contribution of the Perturbation Hamiltonian We are now equipped to evaluate the contribution of *H( . F i r s t we will evaluate the contribution of V( r » then ° f V , j p r a n d f i n a l l y of V . 'The contribution of V should be negligible since i t arises fromty f 4' and contributes only in second order. Similar terns (fourth order in b-fc* ) contribute in f i r s t order the term of order 10 that of the second order term in » and one would expect Vy. to make an even 31 smaller contribution. For this reason i t was not explicitly evaluated. -H,pr * however, since i t arises from )f might be expected to contribute an amount similar to the terms in Xb^p and }ialT arising from ft . The first order contribution vanishes because i t is linear in the phonon operators and . In second order, however, there is a non-vanishing contribution: C- 5 ( lK ,<H ,Prl^)(n)HW t ,prl^> . v s f Ep-Ec . • cur.ifl The intermediate states are of two types describing the fact that a roton of momentum 1\JL will spend part of its life in excited states of momentum , and may do so either by emission and re-absorption of a virtual phonon of momentum -"nK , or by absorption and re-emission of a virtual phonon of momentum t'H K ; thuB (III.17) may be written as L\ r < >[>Vl H* Mt Mr.* /' r- sr t- ' '»r- • m-K i it ' if* K i • i | \ T « | A,,.R| /I II E ^ I Z — : — • ( M- , 8 ) where the summations are restricted not only to l " K ) , |3t|4 K «n»x but also to |3t+Kl^ KW»K since £+K is the propogation vector of the intermediate states of the roton. It was shown in the previous section that the dominant terms in the phonon and roton energies are ()+*)^ C8K a n d ^ \jT respectively; consequently, only these need to be considered in the denominators of (III.18) unless they cancel for some jt and "jt • That they do not cancel can be shown by the following simple calculation. With the above approx-imation one has for the f i r s t term of (III.18) (in. 19) ( 2 Z 7 . 2 / ) where LW..ZZ) = Z.S60 + *) for liquid helium II. The cancelling of the terms in (III.21) requires fl which, since i t is inversely proportional to the fourth power of K m j i r , wi l l happen only i f K mat ie considerably greater than Ziman*s value - a condition that is contrary to the arguments set forth in Chapter II §2.2. Similarly,in the second term of (III.18) one obtains E . - F W « U+4) c . * K + / £ [ll - C M ) 8 ] . i (nr.23) "* "ft T? If the order of summation in (III.18) IB changed so that £« / +K' , K + / 3 i and K=-K', then (III.25) is seen to be just the negative of (III.20); therefore, approximating the phonon and roton energies by their dominant terms is valid in (III.25) as well as in (III.19). With these approximations in the denominators, (III.18) becomes (DT.24) There is a term independent of the number of phonons present, but proportional to the number of rotons; let us consider i t f i r s t . One has for a single roton of momentum'K £. and no phonons present where 7* i s the cosine of the angle between ^ and , and obtains on changing the summation into an integral with the polar axis along £ V > ~ ZZTfy0 1JJ ( ^ K , , , + K + Z£S) ' tM-W When integrated over a sphere of radius Km** this i s just the value obtained by Kaempffer (5). An algebraic error was made in evaluating the integral in his paper and led to the erroneous conclusion that ni,fY gave a larger, but negative, contribution to the energy than no,r • It is necessary to take cognizance of the fact that |K Ki"«* when carrying out the integration in (III.26). The region over which one must integrate in order to satisfy that condition and K4 Kn\»* as well can be pictured geometrically in K -space as follows: Form the solid of revolution consisting of two spheres of radius lfm»x with centers separated X. units. Let the., origin of coordinates be at the center of one of the spheres and be displaced i n the -X direction to the center of the other. Fig. I I I . l shows a section through the axis of Fig. I I I . l . Region of integration of Eq. (III.26). revolution and containing k . Clearly, the integration i n (III.26) ex-tends over the unshaded part of the solid. From symmetry the limits on the f Integral are 0=Oand p--2.1r* . The j 9 and K integrals are straight-forward i f one f i r s t integrates over a sphere of radius K max~-£ , for which the limits on f and K are and then over the remainder for which the limits on f and K. are connected, by For the outer region the integration i s simple, though algebraically some-what tedious, i f the J integral i s carried f i r s t . In this case the limits on and K are After changing to the dimensionless variables (ttZ7j one obtains £ - _ A A 6 (1H.Z8) 7 5 (lH.Z8a) On expanding in a power series in the roton propogation vector and keeping terms to second order in JL t (III.28a) becomes Y<? ttfi)l S t<fl*» V-(III.29) taken together with (III.16) gives the corrected roton energy then as Arm . a \t 29j (nr.30) where, R (UT.3I) It i s evident from (III.51) that contrary to Kaempffer*s note, (III.25) may be considered as a perturbation to the roton energy given by (III.11). (III.25) may be called the "self-energy" of the roton for i t i s independent of the number of phonons and other rotons present. Thus, in order to obtain the contribution o f / " V 1 which i s independent of the number of phonons present, one need only multiply (III.29) by the dis-tribution function M£ and sum over a l l momenta. On the basis of Ziman1s theory the corrected energy (III.30) must be used in the distribution function, and i t is therefore possible to take into account both (III.11) and the "self-energy" by using i t instead of ( I l l . l l ) a n d (III.29). T6 It is evident from (III.JOj that the "self-energy" correction to the roton energy spectrum gives a positive contribution linear in the roton momentum; Landau (12), however, found i t necessary to use a negative linear contribution (or a temperature dependent mass of the roton) in order to obtain second sound velocities in agreement with Peshkov's measurements (16). A positive linear term also causes the specific heat to be much too small and therefore i t appears that Ziman's method is not capable of describing the roton energy in more than a qualitative way. The revised energy proposed by Landau, has the effect of clustering the rotons about the momentum , with £0= I-9Fx 10 cur' . This is somewhat above the upper limit proposed by Ziman. In fact, excitons (phonons or rotons) with the propogation vector To have a wavelength shorter than the average interatomic spacing. This means that the main contribution to the energy of the roton gas is given by rotons with wave lengths less than the average interatomic spacing. Thus i t is entirely possible that a continuous f l u i d model such as Ziman18, with the graininess of the underlying f l u i d entering only by means of a cut-off to the spectrum, w i l l not give the energy i f (III.. 32) is the correct expression. Even i f an expression like (III.32) were ob-tained i t might be d i f f i c u l t to determine the upper limit to the spectrum without a detailed consideration of the underlying f l u i d . The latter con-sideration would be an important one, since Ziman's cut-off l i e s below the maximum in the roton density in Landau's theory. (III.24) has another terra independent of the number of phonons present, but requiring more than one roton; i t is proportional to Mjp the product of two roton distribution functions. However, i t i s negligible as can be shown by noting that the larger expression A 7 7 ( K ' t * ' i ) 2 * ~ » T Mt i n which MJ+tf has been replaced by 6 * T has an energy less than lo" times the "self-energy" below 1°K. So far. we have considered the contribution ofH,|frto the roton energy, the remainder involves both phonon and roton distribution functions. We w i l l now show that this gives a negligible addition to the energy. One has from (III.24), excluding the terms dependent on the roton distribution only, (31.7+) The term in (III.55) of third order in the distribution functions can be shown to be zero by changing the order of summation in the f i r s t term, leaving the energy in the form (111.5^). The number of phonons with energy near the maximum value i s extremely small, and therefore the contribution to Epr w i l l be s i g n i f i -cant only when Similarly, i f the roton energy is given approx-imately by (III.50) the contribution w i l l be significant only when . Because of the Nf becoming extremely small when ft""Ama« , we may extend both sums in (III.J4) to /fw«« . With this approximation Ef>r becomes c ^ ZK+80 K ^ i l y * - 8 ^ * r * K u ~ f 7*> which 16,01) converting the summation over X into an integral, ^ y . f 4 / ? S ^ * * ™ * * ? ] ] ^ .11, (nr. 36) We shall also make the approximation i n the distribution function, and replace X by K m a x in the denominators. The value obtained w i l l then be larger than the actual one. We obtain thus the approximate value Ea = 3 " * -—r—, CJSSKHfc + = C . § (v4 iK + Y*V )Nf l , Iffl-Jfi) K where, 2/ 4 l o " 4 j y - ^ x IO 1 4 I'.'cm.-2- sec* at |*K. Landau and Khalatnikov (13) obtained the value 2.8 x l0"gT lcm; 1 *ecl for Y from other considerations. While (III.38) is negligible as far as a contribution to the 39 energy is concerned, the contribution proportional to K3 is of use in showing that another term obtained later from ^f,fP is also negligible. -It should be noted also that the K3 contribution is positive. This is contrary to the arguments of Landau and Khalatnikov (15) as well as to those put forth by Temperly (17)> We have yet to take into account Hl)f> , corresponding to phonon-phonon interactions. Let us consider the part of third order in the creation and annihilation operators since i t contains the contribution of the terms of order three in the field variables. The terms of other orders in the operators should give a smaller contribution since they arise from higher order terms in the field variables. In particular, the term of second order in the operators, arising from nr and higher order terms, should give a smaller contribution in second order perturbation than in fi r s t order. The latter was taken into account in » giving the term o t f iCoS , and should be comparable to the second order con-tribution of ^ .We will therefore take into account here only the con-tribution of X 0 * . We have from (11.46") The first order contribution vanishes, for i t has an odd number of creation and annihilation operators; the second order contribution, how-ever, does not vanieh}and is given by There are four groups of intermediate states, each containing a sub-group, which we will write as follows! + 0 *> -r , , (^H)(N^ g . * l ) (»V|t . - l )v f 0 ( ^ + , ) ( ^ - 2 ) N K N t f -%{N*+l)l»in+j) (EMU fm.43) (jn .43a) (m H) The sub-groups, IA, Ilk, IIIA and IVA, arise from the special cases in which two or more of the creation and annihilation operators have the same subscript. The denominators corresponding to the intermediate states I, IA, II and IIA are E0-£I-<KW.ir.>-<RiHi«> im.+s) 41 •= n>V(8N2£-4N£ +10), (fll.46a) E0-Eff-- (/+«)c.* C / K + R ' I - K - K ' ) , (nr.47) Eo-F I 4= ^^(-aiv^+^+e). (HI .47a) These are the only cases in which the energy differences in the denomi-nators vanish for certain values of K , K ' *0 when using only the domi-nant terms in yV0)p . As can be seen from (III.46a) and (III.47a), keeping the non-linear terms is necessary to make the denominators non-vanishing. The same could be done in (III.46) and (111.47); i t turns out, however, that the numerator of the energy corresponding to groups I and II vanishes for the saT.e values of K and K1 in such a way that the ratio remains f i n i t e . In fact, the contribution to the energy is small from these values and for this reason the non-linear terms were not included in (111.46) and (111.47). The remaining energy differences are: E0-Em = ( | + « ) C 0 U K * K ' * | K + K ' | ) , (ILT.48) E0- EaA=4(l>n)c.hkt (HT.48a) F0-F17 = -(H-")c 0* ( K + K + lRtKl), CUT.49) Eo -Em = -40t«<)cn k. (nr.49a) 42 The matrix elements are <MH>.)= iffvpW ^ 9 ( ^ ( ^ * 9 • Cm.53a) The perturbation energy Eppcan be divided naturally into four terms, the f i r s t term giving the energy due to a summation over inter-mediate states I and II, the Becond over states III and IV, the third over states IA and IIA and the fourth over states IIIA and IVA. Thus we write Ej>j» =Ei i"E*+El3 +E+ , where E 2 4J cm.5-5) J lip.VH K I -Mg +10 -INjjjtrNjf+S J' E r - s S ^ l ^ W < W ' ) ] . o n . 57) It should be noted that in E< each intermediate state will be counted twice when carrying out the summations, and in £ 2 six times, hence the factors j and in those equations. One would expect and to be much smaller than E, and £ 2 . » _ i 7*1 for the former arise from special values of K and r< ; e.g., for each term of £ 3 there are a large number in E, due to neighboring values of K', so one would expect its value to be less. It is also worth noting that Ej and E4 are independent of the volume; that is, they have a fixed value no matter how large or small the volume of fluid is. can be shown to be small for helium II in al l but extremely . small volumes by noting that, considering only the linear terms, "~20 E v ~ ' (nr. 58) an entirely negligible quantity. f j , however, is more difficult, for distribution functions enter into the denominator as well as into the numerator. If the de-nominator is approximated by its minimum value one obtains - ^ • (nr .5S) 4 4 which is not negligible except for very large volumes. This i s a maximum value since the smallest possible denominator was used. A more detailed evaluation should lead to a smaller energy; however, we w i l l use the following argument instead. It was shown previously that phonon-roton interactions con-tribute a cubic term to the energy of each phonon. The same result, but with a different sign, was obtained by Landau and Chalatnikow (13) from other considerations. Thus i t seems that in actual fact there is a cubic term in the phonon energy. If this term i s taken into account in the de-nominators of intermediate states IA and IIA they become E.-E*" - ( E . - E j = G B 5 C o K s . W.W Since X**IO -*• ID the cubic term i s much larger than the quadratic terms except when K is very small. Using (III.60) i n £3 i t becomes < ( b H ) * 5 KIM *.„ Thu8,as expected, E3 is negligible. The writer is of the opinion that no significance can be attached to the terms independent of the volume for they are only important for extremely small volumes, where one would expect atomistic effects to take over. * Let us consider £, next. For evaluation i t is convenient to change summation variables, assigning toM-*^ in the term linear in the distribution function the single variable K , f$fe obtain then ^ " K * V £ R ' L , * ~ K 1 J ( A ' + I K - A ' I - K ) X K K - K K K K K - K ' K i With the approximation we have used in the denominator i t is zero when and J\> are parallel and K^K'O It will be shown later, how-ever, that using Bose-Einstein statistics the numerator is also zero for these values, and the ratio is finite. Hence, because the term linear in N$ is multiplied by a summation limited only by the maximum allowable K , i t is probably large in comparison with the non-linear terms. The main reason for questioning this is the possibility of a large con-tribution when the denominator is very small; we shall see that this is not the case. Assuming for the moment, then, that the contribution when the denominator is small is not important, we will approximate E, by the term linear in ^ , Since the summation over K' contributes most when K'~K»»* and since N-+ is appreciable only when K « K m a / , one can expand the K u summand in a power series in , and retain only the first few terms. Also, at the upper limit of the T<' summation we must keep 1KLK| ^k^as well as K'<km<(0 We use the approximation |K ' -K | - - K ' O - ^ 5 * ) , (nr.63) where 0 is the angle between K and K' , and obtain *i(b-i)'(i-c..e)K + tHK 5 ))^, (10.64) retaining only terms up to the f i r s t order in ^ ? . K On replacing the summation over K1 by an integral and integrating^ 46 but keeping I K ' - H l i k * ^ as well as K ' $ K«, 4 x ;we obtain for E , E, - - f C ^ I K N - - ^ c , * z k 2 N j , <nr.6yj K K where, S _ (l>V* K^j, , » ( H T . S 6 ) [icb -oNj^Mj* J ( n r . 6 7 ) We go on to e Its denominator never vanishes for K , K ' * O J but, like £, , i t has terms linear in and for the same reasons we ex-pect their contribution to be the principal one (neglecting the zero point energy term)0 After making appropriate changes of summation vari-ables and using approximations similar to (III.63) we obtain for the part linear in Nfc, including terms up to fir s t order in & . K ' E2g"tt«t«fr0v-f 5 [ < M 8 K K > [ f c ( H ) V c M » w ( b - i ) c i - e ^ « ) ] K ' ] N - . (in.68) This becomes on replacing the summation over ft' by an integral and integrating with the restriction that I K + K ' I ; K'< Kmi* K (m.69) in which I is given by (III.66) and *[t by ^ 5a4-rr*(u-«)p6C0 Thus finally the total phonon energy, corrected for phonon-phonon interactions, is in .71) 47 For liquid helium II, with ( b - l ) ~ I, the second order term does not become comparable with the linear term from £ i > p until . This is well above the region in which the summand is appreciable and hence only the linear term needs to be taken into account, leaving P K ^ c( contains the contribution of W 1 4 ' and & the contribution of )f t 3 > to the energy. This corrected energy is applied in the next chapter to the specific heat of liquid helium II. One more point remains! that is to show that the contribution to f. when "K and K 1 in (111.^4) are parallel is small compared to the term linear in . Substitution of (III.8) in (III.54), with F £ linear in K gives 1 I6(i+»0(2ir) p." / / L |K+K'| v e — s , Jj.,,— d'tf <J K **i-g.-l x m-K'-lSrfW (<*H-|) (e**'-t) (e*"<** ' - / ) < m . 7 5 ) where M fcT • This is negative as oan readily be seen by noting that the numerator is never greater than zero. The limiting value of the integrand is finite, for lK*K'|-» K + K' Kt-K'- K'l 1 In order to show that the contribution when T< and K' are nearly parallel is small, let us approximate the integrand of (III.75) by its value when they are parallel, with the further simplification that 48 — = 2.0. K w*x , e - i its maximum value. This gives « - ( i t * / x io" , T' ! E 0 | P for liquid helium II. (flT.76) Unless b is large this is small compared to E a > p ; i t is also of a higher power in T. If the contribution when the denominator is small were large,?/ given by (111.76) should also be large. Since i t is not, we infer that the contribution due to a small denominator is not important. 49 CHAPTER IV POSSIBLE APPLICATION TO LIQUID HELIUM II Since the theory has not taken viscous effects.into account i t is only applicable to non-viscous fluids, of which the only one known is liquid.helium II. Furthermore, i t is only applicable in the region in which phonon effects are predominant; that is, in the region belbv which there is sufficient energy to exite rotational types of motion, which have a rest energy. From the measurements of Kramers, Wasscher and Gorter (8) of the specific heat the phonon contribution predominates below about 0.6°K, since at higher temperatures the specific heat departs from a T3 law. In this Chapter experimental measurements are used to evaluate «( and 6 for helium II. As pointed out by Horton, Frohlich in his theory of super-conductivity introduced a.renormalized velocity of sound. He used a canonical transformation to diagonalize part of the interaction Hamiltonian describing the interactions between electrons and the lattice. It was found then that the observed' velocity of sound could be chosen to make the lattice vibrations essentially harmonic. Thus the effect of interactions with the electrons is interpreted as affecting the observed velocity of sound, rather than leading to anharmonic lattice oscillations. Therefore, by analogy with Frohich's theory of superconductivity i t is necessary to interpret the effect of phonon-phohon interactions as causing a change in the observed velocity of sound rather than the wave-length. Hence with the phonons obeying Bose-Einstein statistics the phonon energy (III.72) leads to a specific heat 50 s 0.0211T^ joules/gm.deg. for helium II, (iS.ld) where V 6 =. ( 1+*-S )c o is the observed velocity of sound at 0°K, which from the measurements' of Atkins and Chase (1) is 237 ~ 2 m./sec. p0 was taken from Keesom (6) as 0.145 gm./cc. •Since*o( and ' & 'enter into the expression for the observed velocity of sound only, i t is not possible to determine the importance of phonon-phohon interactions by measurements of the velocity of sound and the specific heat. If, however, measurements of z, b and could be made, then d-h could be determined from these. Unfortunately, i t is difficult to even estimate the experimental values of them. Also, d and 8 depend on the fourth power of K E a x » so any error in the effective upper limit to the spectrum will be magnified. However, using Ziman1s upper limit one obtains «, = 0 . 2 9 z , S= £ | ( l > - ' ) \ ( I K ) There remains, then, the problem of evaluating b and z from experiments! measurements. Atkins and Stasior (2) have measured first sound velocities at various temperatures and pressures. The relation between pressure and density at the same temperatures has been given by Keesom (6, pages 207 and 240). From these results /p has been plotted as a function of p at 1.25, 1-50 and 1.75°K on Graph IV.1. The values from which the curves have been plotted are given in Table IV.1. (The vapour pressure density hes been diminished by Q.jf> as noted on page 206 of Keesom.) SI TABLE IV.1 p , yr and -~ at various pressures and temperatures. The values ofp are from Keesom (6), those of V from Atkins and Stasior (2). Pressure Atm. T = l.'25° K T - 1.50° »' T = 1.75° K P % • t m v P V w.'iec"' v '. m . s e t : ' t w f g r ' s e c r 2 Vapour Pressure 0 . 1 4 4 8 237 3.88 . 1 4 4 9 235 3 . 8 1 .1450. 233 3 . 7 4 5 .1522 273 4i90 . 1 5 2 4 272 4.85 .1526 2 7 0 4.78 10 . 1 5 8 4 300 5.68 , 1 5 8 5 2 9 9 5 . 6 4 . . .1590 2 9 8 5 . 5 9 -15 . .16^ 6 326 6.50 *1638 325 6 . 4 5 . 1 6 4 5 323 6.34 20 . 1 6 8 1 3 4 6 7.12 . . 1 6 8 5 3 4 5 . 7.06 ' . 1 6 9 4 3 4 2 6.90 • 25 . 1 7 2 2 5<55 7 . 7 4 . 1 7 2 7 562 7 . 5 9 . 1 7 4 1 355 7 . 2 4 It should be noted that the center experimental point does not f a l l on a smooth curve at any of the three temperatures. There does not seem to be any other effect at these points, BO they may.be due to an ex-perimental error. The portions at higher density are near the A point and for thia reason have not been used. Atkins and Stasior claim that their results have an error of about 1%. Keesom makes no statement about the accuracy but quotes p to four significant figures, so the error in i t is probably less than in V, It is now neceBeary to extrapolate.the f i r s t and second de-rivatives to 0°K. In view of the small curvature of the curves, the possible error in the location of the experimental points, and large extrapolation that must be made, the values obtained w i l l be-at best a . crude approximation. It is worth noting, though, that from the graph i t appears that the curvature i s decreasing as the temperature decreases, and therefore that the second derivative may be positive near absolute zero. . 5 2 The derivatives have been evaluated at p = 0.145 gm./pc. at each temperature using Lagrange's interpolation formula. For reasons stated above only the points at p~ 0.14.5* 0.152 and 0.164 gm./cc. were used. Using Newtoh's interpolation formula one obtains at T = 0°K, p =• Q.145 gm./cc. i' • . * Z-~0±100, b'= 4 ±'lo. ( J L X 3 ) i t best these values are approximate limits to the actual one's. For perturbation theory to be valid Z-%1 . - \ Taking b = 4 one obtains '/ ' *= Hi • this is too large for perturbation theory to be valid, but in view of the uncertainty in b, z and K^^' i t is entirely possible that i t is of the order of one or less, which would make the perturbation theory valid. It should be noted that since *yj, is not known derivatives of ^ have been used instead. This approximation may also have intro-duced an error. . • •'. It would be interesting to review this when measurements from which b and z can be determined have been refined to the extent that even small values of them could be ascertained. Measurements of b, z and,, d and & would provide a means of estimating the effective upper limit to the phonon spectrum. 53 APPENDIX A Proof of Eg. (11.33) There enters into the terms of order s and a = 4 i n the Hamiltonian, the expression in which I may be zero, and i t is required that i t be given in the form of a sum of terms in each of which the annihilation operators are to the right of the creation operators. This i s done with the aid of the commutation relations (11.17). Let us consider f i r s t the case of s even, say S = 2p, and p evaluate (A.l) (which we shall c a l l R in this case) for small values of p. We choose for s = p = 0 (A.l) equal to unity, since this extension allows the same form of expression to be used multiplying the roton part of Jf^as multiplies i t for higher powers of s. For p = 0 and p = 1 we obtain then, with the aid of (11.17), But the second and third terms may be combined by interchanging the subscripts of one of them, say the third term, to give Similarly, for p =* 2 we obtain 54 -4VVV-*« "IStStX p It is evident that for p 0, 1, 2 H may be written in the form xfw'(t») a ...» X Since (A.6) holds for these values of p let us assume that i t i s true for a l l p, then we have R M - 5 | : . . I 8*-- - fttT^Zi} W8K -If x X x (ai ) («£ - f l . ). (A.7) In order to simplify (A.7), we make use of the following relation de-veloped with the aid of the commutation relational KJt*"'^t« *t*3>"" KZ? +i ^p-*; -"^ K» K*9"-- '^if-rt-l KZ/>-H f r V J? **** • • • f l f a-tf ' (A-V 5? (A.8) was put in i t s f i n a l form by changing the order of terms in the summation in the same way as in (A.4). Using (A.8) one obtains x -r4> (A.9) But from (A.6) we have An equation for follows by equating the coefficients of equal terms in the d.2'3. We obtain in this way the partial difference equation with boundary conditions We have, then, that i f (A.6) is true for Rf i t is true for 8'*' , and therefore since i t is true for p » 0, 1, 2, 5 i t follows by induction that i t is true for a l l p. Using (A.8) we obtain for s odd, say s e 2p + 1, 56 Kip *2p*» *2p>» K,«t n I lK,rt|+...+K8f+ l T K, -Ki Kz -Kz 2^p+l 2p+l "ini V^S*' * ' I " » ' ~ V ' K * . . " * a . * ' " *• *Z 1 ? - . ^ , (A. 11) and (A. 12) may be used to evaluate f ^ ^ . For p and q up to five they are given in Table A . l . TABLE A.l Tabulation of f . p f f c 5 fP,4 0 1 0 0 0 0 0 1 -1 1 0 0 0 0 2 3 -6 1 0 0 0 3 -15 45 -15 1 0 0 4 105 -420 210 -28 1 0 5 -9*5 4725 -5150 630 -45 1 I REFERENCES 1. Atkins, K. R. and Chase, C. E, Proc. Phys. Soc. (London) A 64 t 826. 1951. 2. Atkins, K. R. and Stasior, R. A. Can. J. Phys. 3 I 1 I I 5 6 . 1955. 5 . Horie, C and Osaka, Y Science Reports of the Tohoku University 28 : 179. 1955. 4. Ito, H. Prog, of Theoret. Phys. 9 « 117. 1955. 5 . Kaempffer, F. A. Can. J. Phys. 32 : 264. 1954. 6. Keesom, W. H. Helium. Elsevier Publishing Co. Inc., Amsterdam. 1952. 7. Kramers, H. A. Physics 18 1 653. 1952 8 . Kramers, H. C., Wasscher, J. D., and Gorter, C. J. Physics 18 « 329. 1952. 9 . Kronig, R. and Thellung, A. Physica 18 : 749. 1952. 10. Lamb, H. Hydrodynamics, 6th ed. Cambridge University Press. 1952. 11. Landau, L. J. Phys. U.S.S.R. 5 1 71. 1941. 12. Landau, L. J. Phys. U.S.S.R. 11 : 9 1 . 1947 13. Landau, L. and Khalatnikov, I. M. J. Exptl. and Theoret. Phys. U.S.S.R. 19 * 637, 709. 1949. 14. Lokken, J. E. Can. J. Phys. 32 i 359. 195*. 15. London, F. Revs. Modern Phys. 17 * 310. 1945. 16. Peshkov, V. P. J. Phys. U.S.S.R. 10 1 589. 1946. 17. Temperley, H. N. V. Proc. Phys. Soc. (London) A 65 1 490. 1952. 18. Thellung, A. Physica 19 : 217. 1955. 19. Wentzel, G. Quantum Theory of Fields, Interscience Publishers, Inc., New York. •• I 9 4 9 . 2 0 . Ziman, J. M. Proc. Roy. Soc. (London) A 219 « 257. 1955. 21. Frohlich, H. Phys. Rev. 79 : 845. 1950. 22. Frohlich, H. Proc. Roy. Soc. (London) A 215 : 291. 1952. 23. Hortoh, 0. K. Private communication to Dr. F. A. Kaempffer. 1955. P H O N O N - P H O N O N I N T E R A C T I O N S I N T H E T H E O R Y OF F L U I D S by JOHN ERWIN LOKKEN A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in PHYSICS We accept this thesis as conforming to the standard required from candidates for the degree of DOCTOR OF PHILOSOPHY Members of the Department of Physics T H E U N I V E R S I T Y OF B R I T I S H C O L U M B I A OCTOBER, 1955 ABSTRACT This thesis is devoted to the effect of phonon-phonon inter-actions on the energy of a non-viscous fluid, and hence to its specific heat. It extends the work of previous authors by taking into account the terras in the Hamiltonian of higher than the second order (lowest order) in the field variables. It is shown that the term of third order in the field variables, contributing in second order, and the term of fourth order, contributing in first order, may give significant con-tributions i f the theory is applied to liquid helium II. The phonon energy in this approximation is linear in the momentum, E = ll +«-e)c.*K. Here, * C04i K is the contribution of the fourth order term and -Sc«,'nK the contribution of the third order term. C„nK is the value obtained by previous authors by considering the lowest order term only. It is shown that for liquid helium II the contribution to the energy of the non-linear terms is smaller than that of the linear terms. In this expression for the energy (I + o( - S)Q, is interpreted as the measured velocity of first sound. Thus the cubic term in the specific heat is unchanged in this approach, the effect of the higher order terms in the Hamiltonian being to change the velocity of sound. Because the non-linear terms have been found to be small the conclusion has been reached that the terms of higher order in the tem-perature than the third cannot be attributed to phonons, and that there-fore this theory is only valid for liquid helium II below 0.6°K. ACKNOWLEDGEMENTS The research described i n the thesis was carried out with the aid of National Research Council Studentships (1952-53* 1953-54) and Summer Scholarships (1953» 1954) to the author. I wish to express my gratitude to Professor F. A. Kaempffer, who supervised this research, for suggesting the problem, for his continued interest in i t and for the many helpful discussions we have had. Finally, I wish to express my appreciation to my wife for her constant encouragement, and the shouldering of many of the parental responsibilities that should have been mine. TABLE OF CONTENTS CHAPTERS PAGE I Introduction and Summary 1 1.1 Historical Review 1' 1.2 Program of This Work 5 II Development of the Hamiltonian i n Operator Form . . . . 9 2.1 The Classical Hamiltonian 9 2.2 Quantization . . 10 2.3 Division into Unperturbed and Perturbation Hamiltonians 19 III The Energy Spectrum of Phonons and Rotons 24 3.1 The State Function . . . 24 3.2 Contribution of the Unperturbed Hamiltonian . . . . . 27 3.3 Contribution of the Perturbation Hamiltonian. . . . 30 IV Possible Applic.ition to Liquid Helium II 49 APPENDIX A Proof of Eq. (II.33) 53 TABLES IV. 1 p f Vand^jj at Various Pressures and Temperatures . , 51 A.l Tabulation of f n 56 TABLE OF CONTENTS (continued) PAGE FIGURE I I I . l Region of Integration of Eq. (III.26) 33 GRAPH FACING PAGE IV. 1 as a Function of p at Various Temperatures . . . . 50 PAGE REFERENCES 57 1 CHAFTER I • ' INTRODUCTION AND SUMMARY Phonon-phonon interactions have been investigated in this thesis as a prototype of a non-linear f i e l d theory, and for their possible applic-ation to the theory of.liquid helium II. The work forms an extension of the quantum hydrodynamics of Kronig and Thellung (9) and Ziman (20) by taking into account higher order terms in the Hamiltonian. 1.1 Historical Review Quantum hydrodynamics is really a f i e l d theoretical approach to the many particle behaviour of a f l u i d , and as such is applicable only when the excitations of the f l u i d - or excitons as H.A. Kramers has called them (7) - have a wave length which is long compared to the interatomic spacing. Although a rigorous formulation was not found until 1952, the idea of quantum hydrodynamics was introduced in a now famous paper by Landau in 19^ 1 (11). It was in connection with liquid helium II that he f i r s t attempted to construct a quantum theory of liquids by direct quant-ization of hydrodynamical variables such as the density, the current and the velocity, without explicit reference to interatomic forces. Thus he sought to express the dynamical behaviour of the liquid at temperatures between zero and a few degrees Kelvin in terms of quantized excitations of the motion, and to this end postulated certain commutation relations between the density, the current and the velocity of the microscopic liquid*; these relations reduce in the macroscopic limit to the ordinary ~ L o n d o n (15) has questioned the representation of the velocity by a linear operator; however, Thellung (18) has shown that except for a minus sign a l l of Landau's commutation rules follow from the more rigor-ous treatment developed independently by Ito (4), Thellung (18) and Ziman (20). 2 hydrodynamical equations in operator form. He assumed that the Hamilton-ian density was given by the sum of the kinetic energy and the internal energy per unit volume. The longitudinal excitations, due to potential motion of the liquid, called "phonons", were found to have an energy proportional to the momentum; that i s , E R = c0p , c0 being the velocity of f i r s t sound, K the propogation constant and p=-fK the momentum of the phonon. This leads to a specific heat proportional to T J. While this is of the correct power In T at low enough temperatures, i t has been shown by H.C. Kramers, Wasscher and Gorter (8) to be too small in the case of liquid helium II. Except for the special case of irrotational motion, the compon-ents of the velocity do not commute, the commutator of the components of the velocity containing a component of the vorticity. Landau then assumed from the analogy with angular momentum** that the vortex motion of the fl u i d must be quantized. Furthermore, for the existence of super-f l u i d i t y the ground state of the vortex motion must l i e above that of the potential motion; hence he conjectured that the energy of en excitation of the vortex spectrum, which he called a "roton", measured on the energy scale on which potential motion is reckoned as zero, may be given by an expression of the form where the effective mass of the roton JJ and the rest energy A are constants ** The analogy is not complete, however, since for a complete analogy the commutator would equal a component of the velocity rather than the vorticity. adjusted to f i t the experimental specific heat curve: This completed a phenomenological description of liquid helium II ex-plaining the experimental results to that time. Later, however, Peahlcov's (16) measurements of second sound velocities indicated a marked increase in the value of// as the temper-ature decreases, whereupon Landau modified the roton spectrum so that the rotons were clustered about one value of momentum, p 0 ; i.e. where now In the same paper he abandonned his previous concept of rotons and instead considered them as being longitudinal excitations of short wave length. Peshkov's measurements were then used to plot an energy spectrum for longitudinal excitations that was nearly linear in the mom-entum for small momenta, corresponding to phonons, but quadratic near Pa corresponding to rotons. In this way agreement with experiment was ob-tained both for specific heat and for second sound velocities. On the assumption that phonons and rotons are the same type of excitation but of different wave length,Landau and Khalatnikov (13) showed that the energy of a phonon of small momentum is of the form Eg = c 0 ( p - yp 5 ) , where Y=2.8 x IO*7 g:1 cm7lsec* . Temperly (17, §9 page 505)> however, has pointed out that this value of X is somewhat less than one would expect from other considerations. Landau and Khalatnikov in the same paper calculated the scatter-ing cross sections for interactions between excitons in connection with the evaluation of the coefficient of viscosity of liquid helium II. The Hamiltonian proposed by Landau (11) contains terms from which the cross section for the scattering of phonons by phonons was computed; in order to keep the cross sections f i n i t e they used the non-linear energy spectrum mentioned in the previous paragraph. Since at that time there was no quantum mechanical Hamiltonian known that included vortex motion, the rotons were considered as small spheres for the purpose of evaluating the cross sections for the scattering of phonons by rotons and rotons by rotons. The coefficient of viscosity calculated using these cross sections, and thus attributable to the excitons, was in good agreement with experi-ment. A great advance in quantum hydrodynamics was made in 1952 by Kronig and Thellung (9) when they considered the irrotational motion of a non-viscous f l u i d from a f i e l d theoretical point of view. They obtained the classical Hamiltonian, which was of the form assumed by Landau in 19^1, starting from Bernoulli's equation and the equation of continuity. Follow-ing Landau they expanded the Hamiltonian in a power series in the canon-i c a l f i e l d variables p (velocity potential) and p (density of the liquid) and then quantized UBing the usual f i e l d theoretical methods. The term of lowest order in the variables (second order) leads to the T term i n the specific heat previously obtained by Landau. No attempt was made to evaluate the contribution from higher order terms. Sometime later Thellung (18), Ito (4) and Ziman (20) independ-ently derived a classical Hamiltonian describing both irrotational and rotational motion of an ideal non-viscous f l u i d . On expanding the Hamiltonian in terms of increasing order in the f i e l d variables the phonon terms previously obtained by Kronig and Thellung result, as well as terms containing only the roton variables and terras containing both phonon and roton variables. Only Ziman was able to obtain any eigen values; and at that, he found them rigorously only for states with no rotons and states with one roton, neglecting the contribution from phonon-roton interactions. In order to keep the energy f i n i t e he found i t necessary to take into account the "graininess" of the underlying f l u i d by the introduction of an upper limit to the propogation vectors. The energy spectrum obtained then was of the form postulated by Landau in 1941, with For purposes of visualization of the roton states in terms of observable movements of the f l u i d , Ziman used cylindrical coordinates and obtained axially symetric solutions to the wave equation. He found, as expected, that the vortex motion is quantized and that a roton corres-ponds to what one would c a l l , classically, a steady rotational motion of the f l u i d , capable of moving as a "vortex" through the volume. This motion, however, since i t is quantized can not persist as a steady state with the vorticity a r b i t r a r i l y small but non-zero everywhere. 1.2 Program of This Work It was the purpose of the work reported in this thesis to extend the theory of Kronig and Thellung (9) by taking into account the inter-, action terms of the general Hamiltonian as set forth by Ziman ( 2 0 ) . For this investigation the Hamiltonian was expressed i n terms of annihilation and creation operators, and arranged into terms of ascending order in the operators, with the annihilation operators operating before the creation operators. In order to get. a system with 3N degrees of freedom, i t was necessary to introduce the "graininess" of the underlying fluid. The ad-vantage of this procedure is that i t introduces an upper limit to the spec-trum, without which the contribution to the energy of terms of order three and higher in the field variables would be infinite. The Hamiltonian in operator form is divided into two parts: the terms that contribute to the energy in first order only, which form the un-perturbed Hamiltonian, )V0 , and the terms that contribute in a l l other orders, which form the perturbation Hamiltonian, . It is shown in Chap-ter III that the unperturbed Hamiltonian gives a contribution to the energy of the phonons which is predominantly linear in the momentum. The major linear contribution comes from the terms of order two in the field variables but there is also a first order contribution from the terms of order four, six, etc. The phonon-phonon interactions described by the perturbation Hamiltonian give a second order contribution linear in the momentum and negative. The main contribution to the energy from phonon-phonon inter-actions arises from the terms of order three and four in the field vari-ables, under the assumption that the series for the internal energy per unit volume converges sufficiently fast that only the first three terms need be considered. It was previously pointed out by the author (14) that the term of order four in the field variables gives a linear contribution, to the phonon energy and could explain the discrepancy between the experi-mental (8) and theoretical specific heat of liquid helium II below 0.6°K. It has since been pointed out by Horton'(23), however, that in view of Frohlich's theory of superconductivity (21,22) the perturbations do not affect the observed specific heat, but rather the quantity to be interpreted as the velocity of sound. Phonon-roton interactions make a negligible contribution to the phonon energy. They give rise to a cubic term in the phonon energy, but i t is of the wrong sign to agree with Landau; i.e., i t is positive whereas Landau and Khalatnikov obtained a negative cubic term using Landau's anergy spectrum for excitons (12). Moreover, the magnitude of )f is less, being of the order 10 3* g~Z cm."2, sec.2, for helium II at 1°K compared, to 2.8 x 1037g.~* cnu2 sec.2, from Landau's theory. They also contribute. ? a linear term to the roton energy of the wrong sign to agree with experi-ment. The theory has been applied in Chapter IV to the case of liquid helium II. The linear terms in the phonon momentum give rise to a T^ term in the specific heat. It has been assumed that the only significant, con-tribution to the linear term in the energy - and thus to the T^ term in the specific heat - is from terms of fourth order and less in the field variables. Unfortunately i t is not yet possible to decide experimentally •whether or not this assumption is justified, for the experimental results from which the coefficients of the- third and fourth order phonon terms, can be calculated do not extend below 1°K. However, higher order terms may be taken into account quite easily i f i t should be necessary since they are included in the theoretical expression. Extension of the experi-mental values below 1°K would be very desirable. The non-linear terms in the energy contribute terms of higher power than the third in the temperature to the specific heatj however, these may be neglected in comparison with the third order term. 8 The conclusion has been reached that, the perturbation terms affect the observed velocity of sound and that therefore they do not account for any discrepancy in the measured T^ term in the specific heat. The terms of higher order i n the temperature are too small to be of im-portance. It would appear that while Ziman1s theory gives a f i r s t approx-imation to the roton energy spectrum, the higher order corrections have the wrong sign*. That the proper corrections are not obtained, however, is perhaps to be expected since the momentum p 0 about which the rotons cluster in Landau's 1947 theory corresponds to a phonon wave length less than the average interatomic spacing. One therefore would hardly expect a theory based on a continuous f l u i d , and which takes the "graininess" of the f l u i d into eccount only through a cut off in the spectrum, to give such an energy spectrum except through a fortuitous accident. It there-fore seems unlikely that.a theory of this type w i l l be adequate at temper-atures above about 0.6°K, where the specific heat departs from a T^ form. Therefore the emphasis i n this work has been on the phonon spectrum, which should give reliable results for temperatures below 0.6°K in the case of liquid helium II. *Recent work by Horie and Osaka (3) shows that phonon-roton inter-actions contribute essentially to the roton spectrum, and that Ziman's perturbation approach i s incorrect. The phonon spectrum is not affected by this. 9 CHAPTER II .DEVELOPMENT OF THE HAMILTONIAN IN OPERATOR FORM In this chapter we discuss the Hamiltonian that w i l l be used in the following chapter to evaluate the energy. The f i r s t section and most of the second is a review of work carried out principally by Ziman (20), and is included for completeness. The division of the Hamiltonian into unperturbed and perturbation parts in the manner described is due to the author, as is the operator expression obtained for the general term of order s in the f i e l d variables. The writer has shown that the roton Hamiltonian is infinite i f a l l the terms are taken into account, and therefore only the one present in an incompressible f l u i d , and classically the moat important, is retained. 2.1 The Classical Hamiltonian .Ziman (20), Thellung (18) and Ito (A) have shown that the classical motion of a f l u i d , including both rotational and irrotational motion, can be described by the Hamiltonian"^ where the integral extends over the volume V of the f l u i d , and the Hamiltonian density H i s given by (D.2) where # The notation of Ziman w i l l be used throughout this thesis except where otherwise noted. 10 i s the compressional energy per unit mass of the f l u i d , and u = -vp - %v Y (M) is the velocity. <p and p , the velocity potential and density of the f l u i d , are canonical variables describing the irrotational (potential) motion of the f l u i d ; the scalar fields y and <r are canonical variables describing the rotational (vortex) motion of the f l u i d . Equation (II.4) i s related to the usual expression for the velocity by the Clebsch transformation (20, 10 page 248). 2.2 Quantization As usual the transition to quantum mechanics is made by re-quiring that the operators corresponding to the f i e l d variables <p and ys and the corresponding conjugate momenta p and <r obey the com-mutation relations with a l l other commutators being zero. Ziman showed that the Hamiltonian could be simplified by the transformation to the complex f i e l d operator defined by Y - ; ? F ! ^ f . -!<?-r>2. m.« ¥ satisfies the commutation relations [ f l f l f l f ' ) ] = W-r'). The Hamiltonian density obtained thus i s , in Hermitian form, Jl p W(p) can be expanded as (15, 9) j p w i , i . i ^ f / > - / w * * . i ( l p f j . ( f - A r t i i ( $ jHrrf *••••> »•» classical where C* = ^ | is the square of the Avelocity of f i r s t sound; fpr the operator ± = j)'1 a similar series i s obtainedi 0 - i - J - d + ^ f ' z J . (ir.io; This expansion is reasonable because classically at temperatures near absolute zero, and averaging over volumes large compared to the inter-atomic spacing one has always / p -/>ol « f o ; i t is shown in y2.J, however, that application of the series (II.10) appears to lead to a divergency in the quantum mechanical treatment used. Arranging the Hamiltonian density in terms of increasing order in the f i e l d variables f , p , f and f one obtains H-r^th" , where (20) H ( y - - £ Miff *t ,f^J>-/>o)1, (I.ila) ^ J- ( The terms of higher order than the fourth are due to the expansion of 12 pWip) and p~l in power series in p -pa. The terms of the Hamiltonian corresponding to .(11.11) are ob-tained by integrating over the period volume V=Lo. The integration is fac i l i t a t e d by resolving the variables into Fourier components in the , period volume; that i s , new operators are defined by the relations ( n . i y which obey the commutation rules (11.13) with a l l other terms commuting. The sums are to be taken over a l l values of the propogation vector K such that -* 2. Tf •* -» (H.I4-) where ^>l\t ^-o'-rj L 0 l j are the vector dimensions of the period volume, and the m a r e positive or negative integers, but not zero (20). Substituting (11.12) into (11.11) and integrating we obtain the total Hamiltonian H * H<Zi+%tC3)* * ( 4V... ., with (H.i5a) V l W p l p Jo f ^ K i K f t K K 3 K - K - l ? 2 - K 3 + 13 - & b- 1 - - b - 8*. b% b.. + A- b** b* b* J R_>.... R-t + j . v i c y )^ K, K i *3 - K i . K j ^ K 3 ^ - ^ K S K, Kj_ K s 3 (lT.I5ol) We now introduce phonon annihilation and creation operators Q_» and «jby the transformation d a n d a* obey the commutation rules K K with a l l other termB commuting. As Kronig and Thellung have shown this transformation diagonalizes % ^ , and as w i l l be shown later, certain parts of the even order terms of (11.15). On substituting (11.16) into (11.15) we obtain xjk-t*t+...Ks ( a l ) f a ^ - f l t . ) . . .(*.-* s>4-, (ir.md) where h, and are unit vectors in the directions Kt and T a , and b and 2 are defined by It i s convenient for what follows to expand (11.18) and use the commutation rules (II.1J) and (11.17) to interchange the order of the operators so that the annihilation operators operate before the creation 00 <s/(j; \jW ail operators.. One then obtains well ordered expressions for fr t rr and 15 The general ease of M is discussed later. The quantities f , jj and £ in (II.20) are defined by f - l i k i * . (D-20 4 = j $ p ' . 11.25) The summations here would be infinite for an ideal f l u i d ; however, in any real f l u i d excitations with wave lengths of the order of the average spacing between atoms, (-Jj) , w i l l not be propogated without some attenu-ation. In fact, i t is unlikely that a;ny can be propogated with wave lengths much shorter than (•)£)• Thus the summations must have an upper / v \ limit not much greater than that corresponding to a wavelength [fjj . Whether or not there should be a weighting factor, and. i f so what i t s form should be, is not known. Ziman has made the suggestion, similar to that used in the Debye theory of specific heats for solids, that since the JN coordinates of the system of N particles depend only on p , p must have JN com-ponents. With this assumption the upper limit is obtained by equating 1 6 the number points in wave-number space inside a sphere of radius m^ax to JNjthue The corresponding minimum wave-length is (E.Z5) so i t i s of the order of magnitude one would expect. (11.24) gives Kmax = 2 i r ( ^ j » 2 i r ( ^ ) = /.5"7 rIC* cm"' for helium, iUM*) where m is the mass of the particles constituting the f l u i d . (11.26) has been used throughout this work. On substituting (11.26) into (11.21), (11.22) and (11.23) we obtain 7 < . " / . * - • • » = 1.18 for helium, A 2 ( 9 V 3 % -ft = 9.28 x to'' 6 e rgs for helium, (IE.Z7) (H .2 7a) (11.23) (IT.28a) (ZT.29) We go on now to the general term V of order i n the f i e l d variables. Let us define ftP by (11.30) 17 While (Redoes not occur in one term of the summations in a l l terms obtained when multiplying i t out are contained among the terms of For small values of p we have ft4-- / . rso • (131) A 3 *« as « * "if -•"I** "i 4* + «•* At %\ - <YVs+ From (II.31) i t would appear that ft could be written in the form A rigorous proof of this without the restriction that the have the same subscript has been relegated to Appendix A. After the calculations carried out in the Appendix one obtains the more general result -ZT...I K . t t i (E.33) and «» K l p + 1 * » K' + Kl +-'- + Klf+' J=» "'••=0 . "ipJ-^Zp-H K> * 2 K 2 ^ - r + ' _ t < 2 j . - » - + 2 " K 2 £ + l 1 8 with being given by the partial difference equation subject to the boundary conditions 'o,0= I, i f fr<0,cpP. (06) Limited attempts have been made to find a solution of (11.55) in closed form but without success. Since the only important use of ff)f is for small values of p with q = 0, 1, a table built up using the dif f e r -ence equation i s adequate and therefore serious attempts at obtaining a solution were not f e l t to be justifiable. From Table A.l page 56 for small values of p and q, i t is evident that • c - * n / ' W ) ! rp>i ~ ~Prp>o - 2 p(p-n.! ' for p up to five at least. But since (11.37) is consistent with (11.55) and (11.56) i t is l i k e l y valid for a l l p. On substituting (11.53) and (11.54) into (II.18d) we obtain for the term of order s in the Hamiltonian 19 1 }l W ^ « , V ' J M V 1 Z Kip-t ' -«jt4$...«5 fl * . . . A * , ) $s2p*i»5r. CU.20J) K, K2 K i J - » * -Kty-r+i ' K Z J . + I ' . 2.3 Division Into Unperturbed and Perturbation Kamlltonians Let us look In more detail at V w t (II.20c). The state function zo discussed in §3.1 (Chapter III) diagonalizes the zero point energy tern (the term independent of the 4$ and d£ ), and as well, the second order term in the ^ i n v o l v i n g one annihilation and one creation operator. The remaining terms of second order in the s a r e not diagonalized by (II.38), but there are terms of second order in the bp ^ that are diago-nal ized. In addition we note that there are terms of fourth order in the a^is and in the b- >$ that are diagonal, viz., ag.O^.a^ , ^ f ^ - ^ f The remaining terms do not contribute in f i r s t order with the state function (II.38). The terms contributing to the energy in f i r s t order do not con-tribute in any other order. Because there are terms which contribute to the energy in f i r s t order exclusively with the remainder not contributing i n f i r s t order, the total Hamiltonian has been divided into an unperturbed Hamiltonian, , and a perturbation Hamiltonian,)/( . }/o contain terms of every even order in the annihilation and creation operators, and hence from every even ordered term in the field'variables. The perturbation Hamiltonian, "X^ , may contain terms of every order in the operators except zero order. For convenience i n writing, "Me and V, are divided into groups of terms involving phonon operators only, terms involving roton operators only and mixed,or interaction,terms involving both phonon and roton oper-ators. These are labelled with the subscripts p, r and pr respectively, thus, (D 39) A superscript i s used to indicate the power in the f i e l d variables to which the term belongs; for instance, J^*? specifies the terms of the un-perturbed Hamiltonian arising from the fourth order term in the f i e l d 21 variables, » a"d involving the phonon operators only. The part of the Hamiltonian (11.20) involving roton operators only i s given by x [ i n v i t o w \\ + *•«, s v u ^ V j -(ff.40) Setting p - 2 - p and changing suffixes V R becomes By using (11.21), (11.22), (II.25) and (11.57) (II.41) can be written i n the form . • s ^ ? r: ot.44) This expression, however, is i n f i n i t e , for the ratio of the (p + l ) t h to the pth term is which for. y>o and large enough p is greater than unity, i s certainly 22 greater than zero for there ie propogation of sound in a l l real fluids. The i n f i n i t y is due to the divergence of the expression for p"' in the quantum mechanical treatment used. Each term in the summation i 2 P is of the form of a phonon zero point energy term (independent of the phonon distribution). It does not seem likely to the writer that the • summations over K interact in such a way that the expansion of p'y converges, and therefore i t appears that i t s expansion i s not valid in the treatment used. As a result of the divergence of p~' we have resorted in this thesis to the use of the f i r s t term in the expansion, p~a , classically the most important. The terms of the Hamiltonian are, taking into account only the f i r s t term, , in the expansion of , t.,, > '(n.4f) •f-= 0 K 2? 2 « where, b, S H)S+* J ^ ' / d , w C\ p » " l p (rr.49) 24 CHAPTER III THE ENERGY SPECTRUM OF PHONONS AND ROTONS In this chapter the energy of an ideal, non-viscous f l u i d is found in terms of phonon and roton gases. It is primarily concerned with the contribution of and \o the phonon energy. To do this i t is necessary to give some consideration to phonon-roton interactions, as well as to phonon-phonon interactions. The contribution of Jl enters through the contribution of through *Hy> and t&tff gives a "self-energy" like contribution to the roton energy. This contribution changes the value given by the unperturbed roton Hamiltonian in the opposite way to that required to give agreement with experiment, thus i t does not appear to be valid.* It i s further pointed out that one would not expect this type of theory to give Landau's re-vised roton energy.3(^r also contributes a cubic term to the energy of each phonon, but i t is much smaller than the linear term from -^p. contributes a negative linear term to the phonon energy. This, together with the contribution from (contained in dftf ), changes the magnitude of the phonon energy. It i s shown that for helium II the non-linear terms are not important. J . l The State Function The lowest order term in the Hamiltonian contains phonon var-iables only and is of order two in the f i e l d variables and of order zero and two in the creation and annihilation operators. Taken alone i t gives rise to a specific heat of the form which for the case of * See also Horie and Osaka (5). 25 liquid helium II i s nearly the experimental value below about 0.6°K -the coefficient being somewhat larger experimentally. Thus in this region perturbations must be small, or at least i f there are several per-turbation terms the sum of them must be small. Higher order terms in the Hamiltonian arise from the expansion of the equation of state in a power series i n p-p0 and from the expansion of f~' i n a similar power series, series which classically are certainly rapidly convergent at low enough temperatures. One would expect that quantum mechanically the series would also converge rapidly, especially at very low temperatures; however, as pointed out in §2.3» this i s not the case for p"' , which diverges. But i t s divergence i s of the same form', as the zero point energy divergence (only in this case i t multiplies the roton Hamiltonian, thereby making the roton energy infinite) and therefore does not necessarily imply that the expansion of the equation of state i s invalid quantum mechanically. Furthermore, there are coefficients of the form jjr ( ^ p3. *-n the equation of state .terms that may* easily con-tribute to the convergence of the series; i n contrast, the expansion of p~' does not have factors of the form -Jft to aid i t s convergence. The validity of the expansion of the equation of state must be assumed i f perturbation theory i s to be applicable to the present calculation. Because of the assumption that the perturbation i s small, the use of a state function which diagonalizss and the roton part of )^*' of second order i n bj« and kg has been considered adequate for evalu-ating the addition to the energy due to phonon-phonon interactions. 26 Such a state function can be 'described in the following way: Let the state function of iv£ excitons with propogation vector Tc be written as ^« . i f the excitons are non-interacting the usual relations hold for the annihilation and creation operators (19); viz. K _ (d.l) When the excitons do not a l l have the same propogation vector and do not interact the state function may be written as the product (m.2) For two types of excitons that do not interact among themselves nor with each other the state function may be written as the product of the state functions for each type of exciton v - y p f t j v . t a - 3 ) In our case we shall let Njf be the number of phonons with mo-mentum and Mj* be the number of rotons with momentum h i , We shall use ^pN^ f or the state functions of phonons and " t ^ ^ f o r the state functions of rotons. f given by ( I I I . 3 ) is the state function of the unperturbed Hamiltonian developed in Chapter II. For ease in writing ( I I I . 3 ) has been written in Dirac notation for much of what follows; that i s , i t has been written in the form I f ) = IN K-SIY>, . Un.'4) where the product of the , Y M J * i * to be understood unless otherwise stated or made evident in the text. 27 3.2 Contribution of the Unperturbed Hamiltonian We shall designate the eigen value of a term i n the Hamiltonian by the subscripts and superscripts carried by that term. For the unper-turbed phonon energy we hftteirt, = < T ^ f J ? ) . a n . s) Explicitly, The f i r s t term is a constant independent of the number of phonons present and is thus a type of zero point energy, which would be in f i n i t e i f there were no upper limit to the phonon spectrum. Henceforth i t w i l l be l e f t out of the equations since i t has a fixed value. The remaining terms are functions of the phonon distribution. We assume in analogy with black body radiation that the phonons obey Bose-Einstein s t a t i s t i c s . This choice is further substantiated by the fact that in solids, for which the Debye theory of specific heats is valid, the phonons obey the same st a t i s t i c s . Hence in order to evaluate Eo^ p we replace the summation over K by an integration and the number of phonons by a Bose-Einstein distribution function, thus . r®*"&/$zr >. (nL7j where E"K is the energy of each phonon with momentum between *fi h" and 1I(K + CIK)# Before carrying out the integration, however, i t is necessary to consider the remainder of the Hamiltonian, for i t has terms linear i n N}t which contribute to the energy Ej£ entering into the distribution function. 28 But even though i t i s necessary to consider the remainder of the Hamiltonian before carrying out the integration exactly, i t i s con-venient at this point to show that for most purposes E 0 ) P may be approx-imated by the term linear i n K . As we shall see later this w i l l simplify several calculations. We j u s t i f y this approximation in part by showing that with o(«a, a necessary condition for -3 perturbation theory to be valid, i t i s consistent by virtue of the non-linear terms then being negligible for helium II. With the assumption that the equation of state converges rapidly enough that we need to consider only the f i r s t .term in 4 and i n I ] , we obtain for helium II In order to make an order of magnitude calculation one may approximate • in the distribution function, and obtain K K 1 J'K y . 5 x i o 4 T ^ + z«T8 - O( io' z o) e r S $ | * (HT.9) (m. \6) Since by assumption cf<he f i r s t term in (III.10), and hence the second in (III.6), is the only one we need to consider in this application. However, in the denominator of second order perturbation calculations the linear terms may cancel; the non-linear terms w i l l then be important. In addition to" E^p we require the unperturbed roton energy be-fore considering the perturbation Hamiltonian. I t i s given after a 29 simple calculation by + rf"P'WH) Go) (HL/M since M_ = M i n an isotropic medium. The f i r s t term of (III.11a) with = £ is the result obtained by Ziman for single roton states. (The assumption that |S/|^=M^ does not hold for this case.) Since the complete roton Hamiltonian diverges, (III.11a) is probably at best a crude approximation to the roton energy. Nevertheless, let us assume that the energy of each roton is in reality E r-A'+g. (nr.iz) where and//' are not very different from A *nd i n (III.6a). This assumption is in agreement with Landau's earliest prediction (10 ) as well as with (III.11a), and is for purposes of this thesis a sufficiently accurate approximation since i t s only use w i l l be in making order of magnitude calculations. Because of the large minimum energy, the rotons w i l l obey Boltzman statistics* thus (III.11a) becomes 50 The integrals in ( i l l . 1 J ) are straightforward, giving Remembering that l7fc| and//"//= ^ »w t (III.l4)may be written as approximately V 2TT1 ^ K ' C L + 8 for helium II below 1°K. Clearly, the f i r s t two terms dominate. Since these arise from the f i r s t term of (II. 11a), E9>r can be taken as — f o r liquid helium II, C-Hr.16) except in cases where this may cancel: for instance, when energy differ-ences are taken (III.16) may cancel with a Bimilar expression. 3.5 Contribution of the Perturbation Hamiltonian We are now equipped to evaluate the contribution of }f . F i r s t we w i l l evaluate the contribution of V ( r » then of V l > p r a n d f i n a l l y of V . The contribution of ^ should be negligible since i t arises fromV f^ and contributes only in second, order. Similar terms (fourth order in tj^bf ) contribute- in f i r s t order the term of order 10 that of the second order terra in > &nd one would expect j. to make an even 51 smaller contribution. For this reason i t was not exp l i c i t l y evaluated. -H,pr » however, since i t arises from )r might be expected to contribute an amount similar to the terms in X^ p and ^ 0 > T arising from . The f i r s t order contribution vanishes because i t is linear in the phonon operators a-g and 0.^ . In second order, however, there is a non-vanishing contribution: L ' > P ' " $ " lo-£i " • ' (H. 17) The intermediate states are of two types describing the fact that a roton of momentum 1\JL w i l l spend part of i t s l i f e in excited states of momentum Ufa*) , and may do so either by emission and re-absorption of a virtual phonon of momentum -"fjK , or by absorption and re-emission of a virtual phonon of momentum "t""K K ; thus (III. 17) may be written as |2 l\r° \K* ti* tt* n * * / ± ^12 (m.'8 where the summations are restricted not only to I K I , Km»*but also to U+K.l^ Km** since £"tK is the propogation vector of the intermediate states of the roton. It was shown in the previous section that the dominant terms in the phonon and roton energies are (|+o04>CoK a n d ^ + iJT respectively; consequently, only these.need to be considered in the denominators of (III.18) unless they cancel for some % and ^ . That they do not cancel can be shown by the following simple calculation. With the above approx-imation one has for the f i r s t term of (III.18) J2 -K / f + K where = Z .S60 + *) for liquid helium II. (tt . 2 Z a ) The cancelling of the terms in (III.21) requires fl 4 2 which, since i t i s inversely proportional to the fourth power of K m „ t w i l l happen only i f Km*x ia considerably greater than Ziman's value - a condition that i s contrary to the arguments set forth in Chapter II §2.2. Similarly, in the second term of (III.18) one obtains If the order of summation in (III. 18) is changed BO that /«= , K + / » i and K = -K', then (III.23) is seen to be just the negative of (111.20); therefore, approximating the phonon and roton energies by their dominant terms is valid in (III.23) as well as in (III.19). With these approximations in the denominators, (III.18) becomes There is a term independent of the number of phonons present, but proportional to the number of rotons; let us consider i t f i r s t . One has for a single roton of momentum "fl and no phonons present r _ fry ^ (K2-H2irr)& where f is the cosine of the angle between * and j£ , and obtains on changing the summation into an integral with the polar axis along i r . _ t&L fff (KtEir) 8 tfdrJrdK , 9 A When integrated over a sphere of radius Km** this i s just the value obtained by Kaempffer ( 5 ). An algebraic error was made in evaluating the integral in his paper and led to the erroneous conclusion that ni,^r gave a larger, but negative, contribution to the energy than no,r • It is necessary to take cognizance of the fact that |K when carrying out the integration in (III.2 6 ) . The region over which one must integrate i n order to satisfy that condition and Km«« as well can be pictured geometrically in K -space as follows! Form the solid of revolution consisting of two spheres of radius ifmta with centers separated £ units. Let the origin of coordinates be at the center of one of the spheres and A* be displaced i n the -X direction to the center of the other. Fig. I I I . l showB a section through the axis of Fig. I I I . l . Region of integration of Eq. (III.2 6 ) . revolution and containing k . Clearly, the integration i n (III.26) ex-tends over the unshaded part of the solid. From symmetry the limits on the f integral are 0=0and p--Zft . The j" and K integrals are straight-forward If one f i r s t integrates over a sphere of radius K max""-& , for which the limits on f and K are and then over the remainder for which the limits on f and K are connected. by For the outer region the integration i s simple, though algebraically some-what tedious, i f the f Integral i s carried f i r s t . In this case the limits on J° and K are After changing to the dimensionless variables x " IT > (BZ7) one obtains £ - .AA. li>6 ' zo+*) 2-V +V E -9^'+^ 4 (m.28) 15 - l i t I - -j£~r \ .11- It. . (nr.z«i) On expanding in a power aeries in the roton propogation vector and keeping terms to second order in X , (III.28a) becomes r- - &g \ 3- 4/3 1 fo*- IZ03 l 0 „ 4 J ^£ . ^ 3 ^ 4 l ) £ . 3 (3 j Kt.tm J (III.29) taken together with (III.16) gives the corrected roton energy then as where, f- = , ^ ' r £ . 4 m H e , A - v r o 7 . (nr.31) It i s evident from (III.31) that contrary to Kaempffer1s note, (III.25) may be considered as a perturbation to the roton energy given by (III.11). (III.25) may be called the "self-energy" of the roton for i t i s independent of the number of phonons and other rotons present. Thus, M(33 in order to obtain the contribution of/r,>PT which is independent of the number of phonons present, one need only multiply (III.29) by the dis-tribution function fV£ and sum over a l l momenta. On the basis of Ziman1s theory the corrected energy (III.30) must be used in the distribution function, and i t is therefore possible to take into account both ( I I I . l l ) and the •self-energy" by using i t instead of ( I l l . l l ) a n d (III.29). 7 6 It is evident from (III.50) that the. "self-energy" correction to the roton energy spectrum gives a positive contribution linear in the roton momentum; Landau ( 1 2 ) , however, found i t necessary to use a negative, linear contribution (or a temperature dependent mass of the roton) in order to obtain second sound velocities in agreement with Peshkov'a measurements ( 1 6 ) . A positive linear term also causes the specific heat to be much too small and therefore i t appears that Ziman's method i s not capable of describing the roton energy in more than a qualitative way. The revised energy proposed.by Landau, has the effect of clustering the rotons about the momentum 'ftij , with £6= i-9Sx\0 cur' . This is somewhat above the upper limit proposed by Ziman. In fact, excitons (phonons or rotons) with the propogation. vector to have a wavelength shorter than the average interatomic spacing. This means that the main contribution to the energy of the roton gas is given by rotons with wave lengths less, than, the average interatomic spacing. Thus i t is entirely possible that a continuous f l u i d model such as Ziman'8, with the graininess of the underlying f l u i d entering only by means of a cut-off to the spectrum, w i l l not give the energy i f ( I I I . 2 2 ) is the correct expression. Even i f an expression like ( I I I ; J 2 ) were ob-tained i t might be d i f f i c u l t to determine the upper limit to the spectrum without a detailed consideration of the underlying f l u i d . The latter con-sideration would be an important one, since Ziman'a cut-off l i e s below the maximum in the roton density in Landau's theory. ( I I I . 2 A ) has another terra independent of the number of phonons present, but requiring more, than one roton; i t is proportional to M^Mj -^j** the product of two roton distribution functions. However, i t is negligible as can be shown by noting that the larger expression A 0 - „ ^ in which Mj^ +'jJ has been replaced byfi has an energy less than 1 0 9 times the "self-energy" below 1°K. So far, we have considered the contribution ofH,ifrto the roton energy, the remainder involves both phonon and roton distribution functions. We w i l l now show that this gives a negligible addition to the energy. One has from (III.24), excluding the terms dependent on the roton distribution only, " 8/.vH(K{^ afg[/-tf^ j} + K{(i^to-rg [ « l - ( r +i?flj/ The term in ( I I I . J 3 ) of third order in the distribution functions can be shown to be zero by changing the order of summation in the f i r s t term, leaving the energy in the form ( 1 1 1 . 3 ^ ) . The number of phonons with energy near the maximum value i s extremely small, and therefore the contribution to Epr w i l l be s i g n i f i -cant only when K«-fnt»«. Similarly, i f the roton energy is given approx-imately by (III.3 0 ) the contribution w i l l be significant only when . Because of the /V^ becoming extremely small when ft^ftma* , we may extend both sums in (III.J4) to . With this approximation £>r becomes t p r - ^ V . £ | (fik^-UTf-.+... (•Jff.54) 58 which is, on converting the summation over JL into an integral, 8trVo g K / Max 4i' 3 t^K^+?)V. We shall also make the approximation i n the distribution function, and replace Jt by Kmax in the denominators. The value obtained w i l l then be larger than the actual one. We obtain thus the approximate value • = -2L*L—* ' C * 2 K k u + 4 i » / H r / b < / - 4 f n R K £ , f * (DT.37) lnr.38) where, 2/ = io y * 5 x 10 5 cm. se c. Landau and Khalatnikov (13) obtained the value 2.8xlfl g" c m " 1 f o r r from other considerations. While (III.38) i s negligible as far as a contribution to the 39 energy 1B concerned, the contribution proportional to K' i s of use i n showing that another term obtained l a t e r from Hlf? i s a l s o n e g l i g i b l e . I t should be noted also that the K3 c o n t r i b u t i o n i s p o s i t i v e . This i s contrary to the arguments of Landau and Khalatnikov (12) &a well as to those put f o r t h by Temperly (17)• We.have yet to take into account Hi)f> , corresponding to phonon-phonon i n t e r a c t i o n s . Let us consider the part of t h i r d order i n the c r e s t i o n and. a n n i h i l a t i o n operators since i t contains the c o n t r i b u t i o n of the terms of order three i n the f i e l d v a r i a b l e s . The terms of other orders i n the operators should give a smaller contribution since they a r i s e from higher order terms i n the f i e l d v a r i a b l e s . In p a r t i c u l a r , the term of second order i n the operators, a r i s i n g from fr and higher, order terms, should give a Bmaller contribution i n second order perturbation than i n f i r s t order. The l a t t e r W S B taken into account in H0/P , giving the term °t , and should be comparable to the second order con-es) * t r i b u t i o n of 5+ . We w i l l therefore take into account here only the con-t r i b u t i o n of We have from (11.46) The f i r s t order contribution vanishes, f o r i t has an odd number of cr e a t i o n and a n n i h i l a t i o n operators; the second order contribution, how-ever, does not vanieh }and l a given by C.--..'.-<T.n<L.'m><ftixLrii,.> (1It > V ~ I E»-Ei ' W There are four groups of intermediate states, each containing a sub-group, which we w i l l write as f o l l o w s i + 0 *> -v-(H;-')(w^g' t |)tMii'- |K •>• ^( NK + 0(^- ' ) (NK' + I) SUBMIT 1) (HMD (IE. 42) (111.12a) (IE .43) (IU.43a) ( I E . t t d ) The sub-groups, IA, IIA, IIIA and IVA, arise from the special cases i n which two or more of the creation and annihilation operators have the same subscript. The denominators corresponding to the intermediate states Z, IA, II and IIA are; 41 = u-rK'-itf+ic'O, (nr.46) -(i*«)c9*\7. K.NJ* +2 K(N2*+1) + K(N*-2)1-+ 8kl(Nr t+0CN1j-2) + | < ^ ^ Kf (N^) + ^ ^ N ^ I ) + K 2 ^ : 2 K N - - J ) J = r^VCSN^ - 4 N * +10), (HI.46a) E 0-E n-- (/+-0c* C / K - ^ ' J - K - K ' ) , (nr.47) These are the only cases in which the energy differences in the denoni-natora vanish for certain values of K , K *0 when using only the domi-nant terras in ^ o,p . As can be seen from (III.46a) and (III.47a), keeping the non-linear terms is necessary to make the denominators non-vaniahing. The same could be done in (III.46) and (III.47); i t turns out, however, that the numerator of the energy corresponding to groups I and II vanishes for the same values of K and K in such a way that the ratio remains f i n i t e . In fact, the. contribution to the energy ia small from these values and for this reason the non-linear terms were not included in (III.46) and (III.47). The remaining energy differencea are: E.-Eur = < l + * ) c E U K * K ^ l K + K ' l ) , (ILT.48) Eo- EaA=Ml+<)cJhk, (nr.48a) Eo-Fur = -d+«)c.* (K + K V I R T I H ) , - Cur.4-9) 4Z The matrix elements are: <frl?C lf.>-<f.l'C.I*>* K <fc»c is) -flft]***** J « H * * » H , * B J A^K*'^-^ t * * * 4 - ' ) , en-si The perturbation energy Epp can be divided n a t u r a l l y i n t o four terms, the f i r s t term g i v i n g the energy due to a summation over i n t e r -mediate states I and II, the second over states III and IV, the t h i r d over states IA and IIA and the fourth over states II IA and IVA. Thus we iteEj>p-Ei +El + ' E 3*Ei' , where wr (m.ss) ^ ^ { ' [ W V ^ V K V I ] , Cin.JT) It should be noted that in E< each intermediate state will be counted twice when carrying out the summations, and in E 2 six times, hence the . factors i and £ in those equations. One would expect Bj and B^ to be much smaller than B, and Ez > for the former arise from special values of K and W ; e.g., for each term of £ 3 there are a large number in E, due to neighboring values of K' so one would expect its value to be lees. It is also worth noting that £ 3 and £ 4 are independent of the volume; that is, they have a fixed value no matter how large or small the volume of fluid is. E + can be shown to be small for helium II in a l l but extremely small volumes by noting that, considering only the linear terms, .,2 1 4 6 " 4(I+«V V H K tO~20 r an entirely negligible quantity. , Ej , however, is more difficult, for distribution functions enter into the denominator as well as into the numerator. If the de-nominator is approximated by its minimum value one obtains 4 4 which is not negligible except for very large volumes. This i s a maximum value since the smallest possible denominator was used. A more detailed evaluation should lead to a smaller energy; however, we w i l l use the following argument instead. It was shown previously that phonon-roton interactions con-tribute a cubic term to the energy of each phonon. The same result, but with a different sign, was obtained by Landau and Chalatnikow (1^) from other considerations. Thus i t seems that in actual fact there is a cubic term in the phonon energy. If this term i s taken into account in the de-nominators of intermediate states IA and IIA they become Since X^fO -+ID~^ the cubic term is much larger than the quadratic terms except when K is very small. Using (III .60) i n E3 i t becomes |E_) = £!d£ I [ N , ( N r0 ( N 1 ? + l ) - ( N 1 » + l ) ( N . t 2 ) N - j ] I < ( B H ) ' S K M v ' p tor. 60 Thus,as expected, E3 is negligible. The writer is of the opinion that no significance can be attached to the terms independent of the volume for they are only important for extremely small volumes, where one would expect atomistic effects to take over. Let us consider £j next. For evaluation i t is convenient to change aummation variables, assigning to ^ in the term linear in the — * distribution function, the single variable: K # tsWS obtain tbjsn-ErK>Pav t%[ . J ( K ' 4 - i K - A ' i - K ) . . ; x . ; K K - K K K K k - h ' K . . . v ^ With the approximation We have used in the denominator i t i s zero when f< and TC' are par a l l e l and K^K'„ It w i l l be shown'later> how-ever, that using Bose-Einstein statistics the numerator i s also zero for these values, and the ratio i s f i n i t e 0 Hence, because the term linear in i s multiplied by a summation limited only by the maximum allowable K , i t i s probably large in comparison with the non-linear terms* The main reason for questioning this i s the possibility of a large con-tribution when the denominator i s very small; we shall see.that this i s not the case e Assuming for.the moment, then, that the contribution when the denominator i s small i s not important, we w i l l approximate E| by the term linear i n fv_ 0 Since the summation over K ' contributes most when K'~I<«MX K and since i s appreciable only when K « K m * / , one can expand the K k summand in a power series in ~, , and retain only the f i r s t few terms. Also, at the upper limit of the K*' summation we must keep 1KLK1 ^kw„as well as M'<K»WO We use the approximation |K'-K|= K ' ( i - i c H • . (m . 63 ) where 9 i s the angle between K and K' , and obtain . + i ( M ' ( i - c . . e ) K + 0(i<*).]Ntf>. .-., ( H I . 6 4 ) retaining only terms up to the f i r s t order in . On replacing the summation over K1' by. an integral and integrating, 46 but keeping |K ' -K l iK „ g x as well as H ' ^ K ^ J W obtain for E , K K where* 1 - . ' ( fr- ' j '* IT _|<4 ^ O>-0£ ? . (nr.66) Ws go on to E 2 < Its denominator never vanishes for K,K'.*0J but,like 6, , i t has terras linear in and for the same reasons we ex-pect their contribution to be the principal one (neglecting the zero point energy term). After making appropriate changes of summation var i -ables and using approximations similar to (III„63) we obtain for the part linear i n N/£, including terms up to f i r s t order in £ , c >m-t*ip0v K K, [ This becomes on replacing the summation over y\' by an integral and integrating with the restriction that I K +K ' I ^K'S ^ J , " f i (IE.69) K in which 0 i s given by (III.66) and ^ by _ [ ^ b - l f - l ( b - 0 l ^ , Cul.70) n * ' * 7 8 4 w » ( ! + • « ) f e C e Thus f i n a l l y the total phonon energy, corrected for phonon-phonon interactions, i s (0.71) 47 For liquid helium II, with (•>-1) ~" I , the second order term does not become comparable with the linear term from E,)P u n t i l ,<~KW)-J1 . This i s well above the region in which the summand is appreciable and hence only the linear term needs to be taken into account, leaving F p •=.(l + «c-m-CeZKNSf. m - 7 Z ) c< contains the contribution of W ( 4 ; and S the contribution of V 3 ) to the energy. This corrected energy i s applied in the -next chapter to the specific heat of liquid helium II. One more point remainsi that i s to show that the contribution to f> when ft and K' in (III*5*0 are parallel is small compared to the term linear in . Substitution of (III.8) in (III.54), with linear in K gives . ' I6(I+^)(2TT)'^.2 II1 (K+K'I s C - o ! J'K r| K eta. v , , -r^rr; d K d K erjj.tf." where This i s negative as can readily be seen by noting that the numerator is never greater than zero. The limiting value of .the integrand, is finite,-fpr ;-In order to show that.the contribution when k and T?' are nearly parallel i s small, l e t us approximate the integrand of (III.75) by it s value when they are parallel, with the further simplification that 48 •i e - i i t s maximum value. This gives ' " "i«o**Mi»)V // (e a' K--)fe a V-0 = - ' ( » « • • / X K T ' T ' 5 for liquid helium II. (ILT .76) Unless b i s large this i s small compared to E t > p ; i t i s also of a higher power in T. If the contribution when the denominator i s small were large,E,' given by ( I I I . 7 6 ) should also be large. Since i t i s not, we infer that the contribution due to a small denominator is not important. 49 CHAPTSR IV POSSIBLE APPLICATION TO LIQUID HELIUM II Since the theory has not taken viscous effects into account i t is only applicable.to non-viscous fluids, of which the only one known is liquid helium II. Furthermore, i t is only applicable in the region in which phonon effects are predominant; that is, in the region below which there is sufficient energy to exits rotational types of motion, which have a rest energy. From the measurements of Kramers, Wasscher and Gorter (8) of the specific heat the phonon contribution predominates below - about 0.6°K, since at higher temperatures the specific heat departs from a T3 law. In this Chapter experimental measurements are used to evaluate M. and 6 for helium II. As pointed out by Horton, Frohlich in his theory of super-conductivity introduced a renormalized velocity of sound. He used a canonical transformation to diagonalize part of the interaction Hamiltonian describing the interactions between electrons and the lattice. It was found then that the observed velocity of sound could be chosen to make the lattice vibrations essentially harmonic. Thus the effect of interactions with the electrons is interpreted as affecting the observed velocity of sound, rather than leading to anharmonic lattice oscillations. Therefore, by analogy with Frohich's theory.of superconductivity i t is necessary to interpret the effect of phonon-phonon interactions as causing a change in the observed velocity of sound rather than, the wave-length. Hence with the phonons obeying Eose-Einstein statistics the phonon energy (III.72) leads to a specific heat 50 L v \sp.v*h* s 0.0211T3 joules/gm.deg. for helium II, (.JS.ld) where V» = (l + * - S ) c o is the observed velocity of sound at 0°K, which from the measurements of Atkins and Chase (1) is 237 - 2 m./sec. p0 was taken from Keesom (6) as 0.14-5 gm./cc. Since at and & enter into the expression for the observed velocity of sound only, i t is not possible to determine the importance of phonon-phonon interactions by measurements of the velocity of sound and the specific heat. If, however, measurements of z, b and K m a x could be made, then ©1-6 could be determined from these. Unfortunately, i t is difficult to even estimate the experimental values of them. Also, d and 8 depend on the fourth power of K^x* s o a nY error in the effective upper limit to the spectrum will be magnified. However, using Ziman*s upper limit one obtains There remains, then, the problem of evaluating b end z from experiments! measurements. Atkins and Stasior (2) have measured first sound velocities at various temperatures and pressures. The relation between pressure and density at the same temperatures has been given by Keesom (6, pages 207 and 240). From these results /p has been plotted as a function of p at 1.25, 1.50 and 1.7.5°K on Graph IV.1. The values from which the curves have been plotted are given in Table IV.1. (The vapour pressure density has been diminished by 0.3$ as noted on page 206 of Keesom.) SI TABLE IV.1 p , and — at various pressures and temperatures. The values of p are from Keesom ( 6 ) , those of <? from Atkins and Stasior (2). Pressure Atm. T = 1.25° K T = 1.50° K T = 1 . 7 5 ° K P g. Cm'' X I . s t c " 1 P g.cnr.:1 tr p V m.se<:' vy'xio-* Vapour Pressure 0 . 1 4 4 8 257 3.88 . 1 4 4 9 235 5 . 8 1 .1450 255 3.74 5 .1522 275 4.90 .1524 272 4.85 .1526 270 4.78 10 .1584 300 5.68 .1585 299 5 . 6 4 .1590 298 5.59 15 .I656 326 6.50 .1638 325 6.45 .1645 523 6 . 5 4 20 . 1 6 8 1 3 4 6 7.12 .1685 345 7.06 .1694 3 4 2 6.90 25 .1722 365 7.74 .1727 562 7.59 .1741 3 5 5 7 . 2 4 It should be noted that the center experimental point does not f a l l on a smooth curve at any of the three temperatures. There does not seem to be any other effect at these points, so they may be due to an ex-perimental error. The portions at higher density are near the X point and for this reason have not been used. Atkins and Stasior claim that their results have an error of about 1%. Keesom makes no statement about the accuracy but quotes p to four significant figures, so the error in i t is probably less than i n V . It is now necessary to extrapolate the f i r s t and second de-rivatives to 0°K. In view of the small curvature of the curves, the possible error in the location of the experimental points, and large extrapolation that must be made, the values obtained w i l l be at best a crude approximation. It iB worth noting, though, that from the graph i t appears that the curvature is decreasing as the temperature decreases, and therefore that the second derivative may be positive near absolute zero. 5 2 The derivatives have been evaluated at p - 0.14.5 gm./cc. st each temperature using Lagrange's interpolation formula. For reasons stated above only the points at^«~ 0.145» 0.152 and 0.164 gm./cc. were used. Using Newton's interpolation formula one obtains at T = 0°K, p - 0.145 gm./cc. 2 ~ 0 ± 100, b = 4 ± IO. (H-3) At best these values are approximate limits to the actual ones. For perturbation theory to be valid Z £ I . - Taking b = 4 one obtains This is too large for perturbation theory to be valid, but in view of the uncertainty in b, z and K J B a x i t is entirely possible that i t is of the order of one or less, which would make the perturbation theory valid. It should be noted that since /p is not known derivatives of . ^ have been used instead. This approximation may also have intro-duced an error. It would be interesting to review this when measurements from which b and z can be determined have been refined to the extent that even small values of them could be ascertained.- Measurements of h, z and 4 and £ would provide a means of estimating the effective upper limit to the phonon spectrum. APPENDIX A Proof of Eg. (11.33) There enters into the terms of order s and 3 - A i n the Hamiltonian, the expression in which I may be zero, and i t is required that i t be given in the form of a sum of terms in each of which the annihilation operators are to the right of the creation operators. This is done with the aid of the commutation relations (11.17). Let us consider f i r s t the case of 8 even, say S = 2p, and evaluate (A.l) (which we shall c a l l R in this case) for small values of p. We choose for s = p - 0 (A.l) equal to unity, since this extension allows the same form of expression to be used multiplying the roton part of V^as multiplies i t for higher-powers of s. For p = 0 and p = 1 we obtain then, with the aid of (11.17), But the second and third terms may be combined by interchanging the subscripts of one of them, say the third term, to give . Similarly, for p =• 2 we obtain 54 -4VVV-*« ' " ^ " V ^ -4<*VV-*f It i s evident that for p a 0, 1, 2 RP may be written in the form x ? w T ? ) « » « ; . . . ^ } fl . . . . W ) Since (A.6) holds for these values of p let us assume that i t i s true for a l l p, then we have x («• Hoi - f l . ). (A.7) In order to simplify (A,7), we make use of the following relation de-veloped with the aid of the commutation relational 55 (A.8) was put in i t s f i n a l form by changing the order of terms in the summation in the same way as in (A.4). UBing (A.8) one obtains R P* = II ...l lf„* _ /K,K....K, 2^ Z L L ^ . x But from (A.6) we have An equation for | ^ follows by equating the coefficients of equal terms in the d-jf'j. We obtain in this way the partial difference equation with boundary conditions We have, then, that i f (A.6) is true for R P i t is true for R '* ' , and therefore since i t is true for p « 0, 1, 2, 5 i t follows by induction that i t i s true for a l l p. Using (A.8) we obtain for s odd, say s •= 2p + 1, 5 6 P -at ) = l l l V i ? ^ ^ + K / K » K « - S + ' j * ^ ' V ' S * « ' ' V < K * ^ * r " L (A.11) and (A.12) may be used to evaluate -fp^ . For p and q to five they are given in Table A . l . TABLE A.l Tabulation of f ft* * p f P,. fP,4 0 1 0 0 0 0 0 1 -1 1 0 0 0 0 2 5 -6 1 0 0 0 3 -15 45 -15 1 0 0 4 105 -420 210 -28 1 0 5 -9*5 4725 -5150 650 -45 1 REFERENCES 1. Atkins, K. R. and Chase, C. E. Pfoe. Phys. Soc (London) A 64 i 826. 1951. 2. Atkins, K. R. and Stasior, R. A. Can. J. Phys. 51*1156. 1955. 3. Horie, C and Osaka, Y Science Reports of the Tohoku University 28 i 179. 1955. 4. Ito, H. Prog, of Theoret. Phys. 9 » 117. 1955. 5. Kaempffer, F. A. Can. J. Phys. 52 J 264. 1954. 6. Keesom, W. H. Helium. Elsevier Publishing'Co. Inc., Amsterdam. 1952. 7. Kramers, H. A. Physica 18 i 653. I952 8. Kramers, H. C , Wasscher, J. D., and Gorter, C. J. Physica 18 1 329. 1952. 9. Kronig, R. and Thellung, A. Physica 18 1 749. 1952. 10. Lamb, H. Hydrodynamics, 6th ed. Cambridge University Press. 1952. 11. Landau, L. J. Phys. U.S.S.R. 5 : 71. 1941. 12. Landau, L. J. Phys. U.S.S.R. 11 : 91. 1947 15. Landau, L. and Khalatnikov, I. M. J. Exptl. and Theoret. Phys. U.S.S.R. 19 * 637, 709. 1949. 14. Lokken, J. E. Can. J. Phys. 32 i 359. 1952*. 15. London, F. Revs. Modern Phys. 17 « 310. 1945. 16. Peshkov, V. P. J. Phys. U.S.S.R. 10 : 589. 1946. 17. Temperley, H. N. V. Proc. Phys. Soc. (London) A 65 » 490. 1952. 18. Thellung, A. Physica 19 : 217. 1955. 19. Wentzel, 0. Quantum Theory of Fields, Interscience Publishers, Inc., Hew York. 1949. 20. Ziman, J. M. Proc. Roy. Soc. (London) A 219 « 257. 1955. 21. Frohlich, H. Phys. Rev. 79 : 845. 1950. 22. Frohlich, H. Proc. Roy. Soc. (London) A 215 : 291. 1952. 23. Horton, G. K. Private communication to Dr. F. A. Kaempffer. 1955.
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Phonon-phonon interactions in the theory of fluids Lokken, John Erwin 1955
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Title | Phonon-phonon interactions in the theory of fluids |
Creator |
Lokken, John Erwin |
Publisher | University of British Columbia |
Date Issued | 1955 |
Description | This thesis is devoted to the effect of phonon-phonon interactions on the energy of a non-viscous fluid, and hence to its specific heat. It extends the work of previous authors by taking into account the terms in the Hamiltonian of higher than the second order (lowest order) in the field variables. It is shown that the term of third order in the field variables, contributing in second order, and the term of fourth order, contributing in first order, may give significant contributions if the theory is applied to liquid helium II. The phonon energy in this approximation is linear in the momentum, Ek = (1 + ∝-δ)c₀kk- Here, c₀kk is the contribution of the fourth order term and -δc₀kk the contribution of the third order term. c₀kk is the value obtained by previous authors by considering the lowest order term only. It is shown that for liquid helium II the contribution to the energy of the non-linear terms is smaller than that of the linear terms. In this expression for the energy (1 + ∝ - δ)c₀ is interpreted as the measured velocity of first sound. Thus the cubic term in the specific heat is unchanged in this approach, the effect of the higher order terms in the Hamiltonian being to change the velocity of sound. Because the non-linear terms have been found to be small the conclusion has been reached that the terms of higher order in the temperature than the third cannot be attributed to phonons, and that therefore this theory is only valid for liquid helium II below 0.6°K. |
Subject |
Fluids |
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Thesis/Dissertation |
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Text |
Language | eng |
Date Available | 2012-01-31 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
IsShownAt | 10.14288/1.0085945 |
URI | http://hdl.handle.net/2429/40392 |
Degree |
Doctor of Philosophy - PhD |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
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UBCV |
Scholarly Level | Graduate |
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